# questions

tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
title:
if-lo-g-a-lo-g-b-and-lo-g-c-are-in-ap-and-also-lo-g-a-lo-g-2-b-lo-g-2-b-lo-g-3-c-lo-g-3-c-lo-g-a-are-in-ap-then

slug:
if-lo-g-a-lo-g-b-and-lo-g-c-are-in-ap-and-also-lo-g-a-lo-g-2-b-lo-g-2-b-lo-g-3-c-lo-g-3-c-lo-g-a-are-in-ap-then-49580

question:
<div class="question">If $\log$ a, $\log \mathrm{b}$, and $\log \mathrm{c}$ are in A.P. and also $\log$ a $-\log 2 b, \log 2 b-\log 3 c, \log 3 c-\log$ a are in A.P.,
<br/> <br/>then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{a}, \mathrm{b}, \mathrm{c},$ are in $\mathrm{H.P.}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a, 2 b, 3 c$ are in A.P.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are the sides of a triangle</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">none of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are the sides of a triangle</span> </div>

answer:
<div class="solution">\(\log \mathrm{a}, \log \mathrm{b}, \log \mathrm{c}\) are in A.P.
<br/> <br/>\(\begin{aligned}
<br/> <br/>&amp;\Rightarrow 2 \log \mathrm{b}=\log \mathrm{a}+\log \mathrm{c} \\
<br/> <br/>&amp;\Rightarrow \log \mathrm{b}^{2}=\log (\mathrm{ac}) \\
<br/> <br/>&amp;\Rightarrow \mathrm{b}^{2}=\mathrm{ac} \Rightarrow \mathrm{a}, \mathrm{b}, \mathrm{c} \text { are in G.P. } \\
<br/> <br/>&amp;\log \mathrm{a}-\log 2 \mathrm{~b}, \log 2 \mathrm{~b}-\log 3 \mathrm{c}, \log 3 \mathrm{c}-\log \text { a are in A.P. }
<br/> <br/>\end{aligned}\)
<br/> <br/>\(\Rightarrow 2(\log 2 b-\log 3 c)=(\log a-\log 2 b)+(\log 3 c-\log a)\)
<br/> <br/>\(\Rightarrow 3 \log 2 \mathrm{~b}=3 \log 3 \mathrm{c} \Rightarrow 2 \mathrm{~b}=3 \mathrm{c}\)
<br/> <br/>Now, \(b^{2}=a c \Rightarrow b^{2}=a \cdot \frac{2 b}{3} \Rightarrow b=\frac{2 a}{3}, c=\frac{4 a}{9}\)
<br/> <br/>\(\begin{aligned}
<br/> <br/>&amp;\text { i.e., } a=a, b=\frac{2 a}{3}, c=\frac{4 a}{9} \\
<br/> <br/>&amp;\Rightarrow a: b: c=1: \frac{2}{3}: \frac{4}{9}=9: 6: 4
<br/> <br/>\end{aligned}\)
<br/> <br/>Since, sum of any two is greater than the \(3^{\mathrm{rd}}, \mathrm{a}, \mathrm{b}, \mathrm{c}\) form a triangle.</div>


title: if-m-is-magnetic-moment-and-b-is-the-magnetic-field-then-the-torque-is-given-by-1
slug: if-m-is-magnetic-moment-and-b-is-the-magnetic-field-then-the-torque-is-given-by-1-31828
tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2012']
answer: <div class="solution">$\vec{\tau}=\vec{m} \times \vec{B}$</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $m$ is magnetic moment and $B$ is the magnetic field, then the torque is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\vec{m} \cdot \vec{B}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{|\vec{m}|}{|\vec{B}|}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\vec{m} \times \vec{B}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$|\vec{m}| \cdot|\vec{B}|$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\vec{m} \times \vec{B}$</span> </div>


title: if-m-is-magnetic-moment-and-b-is-the-magnetic-field-then-the-torque-is-given-by
slug: if-m-is-magnetic-moment-and-b-is-the-magnetic-field-then-the-torque-is-given-by-34584
tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $M$ is magnetic moment and $B$ is the magnetic field, then the torque is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\overrightarrow{\mathrm{M}} \cdot \overrightarrow{\mathrm{B}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{|\overrightarrow{\mathrm{M}}|}{|\overrightarrow{\mathrm{B}}|}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\overrightarrow{\mathrm{M}} \times \overrightarrow{\mathrm{B}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$|\overrightarrow{\mathrm{M}} \| \overrightarrow{\mathrm{B}}|$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\overrightarrow{\mathrm{M}} \times \overrightarrow{\mathrm{B}}$</span> </div>

answer:
<div class="solution">Torque, $\tau=\overrightarrow{\mathrm{M}} \times \overrightarrow{\mathrm{B}}$</div>


title: if-ma-x-nb-y-1-touches-the-ellipse-a-2-x-2-b-2-y-2-1-then
slug: if-ma-x-nb-y-1-touches-the-ellipse-a-2-x-2-b-2-y-2-1-then-66146
tags: ['Mathematics', 'Ellipse', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\frac{\mathrm{x}}{\mathrm{ma}}+\frac{\mathrm{y}}{\mathrm{nb}}=1$ touches the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$ then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{m}^{2}=\frac{\mathrm{n}^{2}}{\mathrm{n}^{2}-1}$ or $\mathrm{n}^{2}=\frac{\mathrm{m}^{2}}{\mathrm{~m}^{2}-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{m}^{2}=\frac{\mathrm{n}^{2}}{\mathrm{n}^{2}+1}$ or $\mathrm{n}^{2}=\frac{\mathrm{m}^{2}}{\mathrm{~m}^{2}+1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{m}^{2}=\frac{\mathrm{n}^{2}+1}{\mathrm{n}^{2}}$ or $\mathrm{n}^{2}=\frac{\mathrm{m}^{2}+1}{\mathrm{~m}^{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{m}^{2}=\frac{\mathrm{n}^{2}-1}{\mathrm{n}^{2}}$ or $\mathrm{n}^{2}=\frac{\mathrm{m}^{2}-1}{\mathrm{~m}^{2}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{m}^{2}=\frac{\mathrm{n}^{2}}{\mathrm{n}^{2}-1}$ or $\mathrm{n}^{2}=\frac{\mathrm{m}^{2}}{\mathrm{~m}^{2}-1}$</span> </div>

answer:
<div class="solution">The line is $y=-\frac{\text { nb }}{\text { ma }} x-n b$. It will touch the ellipse if $(-n b)^{2}=a^{2}\left(-\frac{n b}{m a}\right)^{2}+b^{2}$
<br/> <br/>$\left[\mathrm{c}^{2}=\mathrm{a}^{2} \mathrm{~m}^{2}+\mathrm{b}^{2}\right]$
<br/> <br/>$\Rightarrow n^{2}=\frac{n^{2}}{m^{2}}+1$
<br/> <br/>$\Rightarrow \mathrm{m}^{2}=\frac{\mathrm{n}^{2}}{\mathrm{n}^{2}-1}$ or $\mathrm{n}^{2}=\frac{\mathrm{m}^{2}}{\mathrm{~m}^{2}-1}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
title:
if-malphaleftbeginarrayccccos-alpha-sin-alpha-0-sin-alpha-cos-alpha-0-0-0-1endarrayright-mbetaleftbeginarrayccccos-beta-0-sin-beta-0-1-0-sin-beta-0-co

slug:
if-malphaleftbeginarrayccccos-alpha-sin-alpha-0-sin-alpha-cos-alpha-0-0-0-1endarrayright-mbetaleftbeginarrayccccos-beta-0-sin-beta-0-1-0-sin-beta-0-co-33688

question:
<div class="question">If \(M(\alpha)=\left[\begin{array}{ccc}\cos \alpha &amp; -\sin \alpha &amp; 0 \\ \sin \alpha &amp; \cos \alpha &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]\); \(M(\beta)=\left[\begin{array}{ccc}\cos \beta &amp; 0 &amp; \sin \beta \\ 0 &amp; 1 &amp; 0 \\ -\sin \beta &amp; 0 &amp; \cos \beta\end{array}\right]\) then \([M(\alpha) M(\beta)]^{-1}\) is equal to -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{M}(\beta) \mathrm{M}(\alpha)\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\mathrm{M}(-\alpha) \mathrm{M}(-\beta)\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\mathrm{M}(-\beta) \mathrm{M}(-\alpha)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(-\mathrm{M}(\beta) \mathrm{M}(\alpha)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{M}(-\beta) \mathrm{M}(-\alpha)\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \left.[\mathrm{M}(\alpha) \mathrm{M}(\beta)]^{-1}=\mathrm{M}(\beta)^{-1} \mathrm{M}(\alpha)^{-1}\right] \\<br/>&amp; \operatorname{Now~M(\alpha )^{-1}}=\left[\begin{array}{ccc}<br/>\cos \alpha &amp; \sin \alpha &amp; 0 \\<br/>-\sin \alpha &amp; \cos \alpha &amp; 0 \\<br/>0 &amp; 0 &amp; 1<br/>\end{array}\right] \\<br/>&amp; =\left[\begin{array}{ccc}<br/>\cos (-\alpha) &amp; -\sin (-\alpha) &amp; 0 \\<br/>\sin (-\alpha) &amp; \cos (-\alpha) &amp; 0 \\<br/>0 &amp; 0 &amp; 1<br/>\end{array}\right]=\mathrm{M}(-\alpha) \\<br/>&amp; M(\beta)^{-1}=\left[\begin{array}{ccc}<br/>\cos \beta &amp; 0 &amp; -\sin \beta \\<br/>0 &amp; 1 &amp; 0 \\<br/>\sin \beta &amp; 0 &amp; \cos \beta<br/>\end{array}\right] \\<br/>&amp; =\left[\begin{array}{ccc}<br/>\cos (-\beta) &amp; 0 &amp; \sin (-\beta) \\<br/>0 &amp; 1 &amp; 0 \\<br/>-\sin (-\beta) &amp; 0 &amp; \cos (-\beta)<br/>\end{array}\right]=\mathrm{M}(-\beta) \\<br/>&amp; {[\mathrm{M}(\alpha) \mathrm{M}(\beta)]^{-1}=\mathrm{M}(-\beta) \mathrm{M}(-\alpha)} \\<br/>&amp;<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
title:
if-mathrma-0-mathrmb-0-mathrmc-0-are-respectively-the-mathrmpth-mathrmq-th-rth-terms-of-gp-then-the-value-of-the-determinant-leftbeginarrayllllog-math

slug:
if-mathrma-0-mathrmb-0-mathrmc-0-are-respectively-the-mathrmpth-mathrmq-th-rth-terms-of-gp-then-the-value-of-the-determinant-leftbeginarrayllllog-math-74239

question:
<div class="question">If \(\mathrm{a} &gt; 0, \mathrm{~b} &gt; 0, \mathrm{c} &gt; 0\) are respectively the \(\mathrm{pth}, \mathrm{q}\) th, rth terms of GP., then the value of the determinant \(\left|\begin{array}{lll}\log \mathrm{a} &amp; \mathrm{p} &amp; 1 \\ \log \mathrm{b} &amp; \mathrm{q} &amp; 1 \\ \log \mathrm{c} &amp; \mathrm{r} &amp; 1\end{array}\right|\) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">Let \(A\) be the 1 st term and \(\mathrm{R}\) the common ratio of G.P., then<br/>\(\begin{aligned}<br/>&amp; a=T_p=A R^{p-1} \\<br/>&amp; \therefore \log \mathrm{a}=\log \mathrm{A}+(\mathrm{p}-1) \log \mathrm{R}<br/>\end{aligned}\)<br/>Similarly, \(\log \mathrm{b}=\log \mathrm{A}+(\mathrm{q}-1) \log \mathrm{R}\)<br/>and \(\log \mathrm{c}=\log \mathrm{A}+(\mathrm{r}-1) \log \mathrm{R}\)<br/>\(\therefore \Delta=\left|\begin{array}{lll}<br/>\log A+(p-1) \log R &amp; p &amp; 1 \\<br/>\log A+(q-1) \log R &amp; q &amp; 1 \\<br/>\log A+(r-1) \log R &amp; r &amp; 1<br/>\end{array}\right|\)<br/>Split into two determinants and in the first take \(\log A\) common and in the second take \(\log \mathrm{R}\) common<br/>\(\Delta=\log A\left|\begin{array}{lll}<br/>1 &amp; p &amp; 1 \\<br/>1 &amp; q &amp; 1 \\<br/>1 &amp; r &amp; 1<br/>\end{array}\right|+\log R\left|\begin{array}{lll}<br/>p-1 &amp; p &amp; 1 \\<br/>q-1 &amp; q &amp; 1 \\<br/>r-1 &amp; r &amp; 1<br/>\end{array}\right|\)<br/>Apply \(\mathrm{C}_1 \rightarrow \mathrm{C}_2 \rightarrow \mathrm{C}_3\) in the second<br/>\(\Delta=0+\log R\left|\begin{array}{lll}<br/>0 &amp; p &amp; 1 \\<br/>0 &amp; q &amp; 1 \\<br/>0 &amp; r &amp; 1<br/>\end{array}\right|=0\)</div>


tags: ['Mathematics', 'Area Under Curves', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
title:
if-mathrmimathrmmint1mathrmeln-mathrmxmathrmm-mathrmdx-where-mathrmm-in-mathrmn-then-i1010-i9-is-equal-to

slug:
if-mathrmimathrmmint1mathrmeln-mathrmxmathrmm-mathrmdx-where-mathrmm-in-mathrmn-then-i1010-i9-is-equal-to-37206

question:
<div class="question">If \(\mathrm{I}_{\mathrm{m}}=\int_1^{\mathrm{e}}(\ln \mathrm{x})^{\mathrm{m}} \mathrm{dx}\), where \(\mathrm{m} \in \mathrm{N}\), then \(I_{10}+10 I_9\) is equal to -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{e}^{10}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{\mathrm{e}^{10}}{10}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\mathrm{e}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\mathrm{e}-1\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{e}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \mathrm{I}_{10}=\int_1^{\mathrm{e}} 1 \cdot(\ln \mathrm{x})^{10} \mathrm{dx}=\left[(\ln \mathrm{x})^{10} \mathrm{x}\right]_1^{\mathrm{e}} \\<br/>&amp; -\int_1^{\mathrm{e}} 10(\ln \mathrm{x})^9 \cdot \frac{1}{\mathrm{x}} \cdot \mathrm{xdx}=\mathrm{e}-0-10 \int_1^{\mathrm{e}}(\ln \mathrm{x})^9 \mathrm{dx} \\<br/>&amp; =\mathrm{e}-10 \mathrm{I}_9+10 \mathrm{I}_9=\mathrm{e}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">14</span> </div>
source_db: MarksBatch2_P1.db
title:
if-mathrmm-arithmetic-means-are-inserted-between-1-and-31-so-that-the-ratio-of-the-7text-th-and-mathrmm1text-th-means-is-5-9-then-find-the-value-of-m

slug:
if-mathrmm-arithmetic-means-are-inserted-between-1-and-31-so-that-the-ratio-of-the-7text-th-and-mathrmm1text-th-means-is-5-9-then-find-the-value-of-m-80579

question:
<div class="question">If \(\mathrm{m}\) arithmetic means are inserted between 1 and 31 so that the ratio of the \(7^{\text {th }}\) and \((\mathrm{m}-1)^{\text {th }}\) means is \(5: 9\), then find the value of \(m\).</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">14</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">24</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">20</span> </li> </ul>

answer:
<div class="solution">Let the means be \(x_1, x_2, \ldots \ldots x_m\) so that \(1, \mathrm{x}_1, \mathrm{x}_2, \ldots . \mathrm{x}_{\mathrm{m}}, 31\) is an A.P. of \((\mathrm{m}+2)\) terms.<br/>Now, \(31=\mathrm{T}_{\mathrm{m}+2}=\mathrm{a}+(\mathrm{m}+1) \mathrm{d}=1+(\mathrm{m}+1) \mathrm{d}\)<br/>\(\therefore \mathrm{d}=\frac{30}{\mathrm{~m}+1}\) Given : \(\frac{\mathrm{x}_7}{\mathrm{x}_{\mathrm{m}-1}}=\frac{5}{9}\)<br/>\(\begin{aligned}<br/>&amp; \therefore \frac{\mathrm{T}_8}{\mathrm{~T}_{\mathrm{m}}}=\frac{\mathrm{a}+7 \mathrm{~d}}{\mathrm{a}+(\mathrm{m}-1) \mathrm{d}}=\frac{5}{9} \\<br/>&amp; \Rightarrow 9 \mathrm{a}+63 \mathrm{~d}=5 \mathrm{a}+(5 \mathrm{~m}-5) \mathrm{d} \\<br/>&amp; \Rightarrow 4.1=(5 \mathrm{~m}-68) \frac{30}{\mathrm{~m}+1} \\<br/>&amp; \Rightarrow 2 \mathrm{~m}+2=75 \mathrm{~m}-1020 \Rightarrow 73 \mathrm{~m}=1022 \\<br/>&amp; \therefore \quad \mathrm{m}=\frac{1022}{73}=14<br/>\end{aligned}\)</div>


title: if-mathrmpmathrma-cup-mathrmbfrac23-mathrmpmathrma-cap-mathrmbfrac16-and-mathrmpmathrmafrac13-then
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
slug:
if-mathrmpmathrma-cup-mathrmbfrac23-mathrmpmathrma-cap-mathrmbfrac16-and-mathrmpmathrmafrac13-then-27196

question:
<div class="question">If \(\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{2}{3}, \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{6}\) and \(\mathrm{P}(\mathrm{A})=\frac{1}{3}\) then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">A and \(\mathrm{B}\) are independent events</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">A and \(\mathrm{B}\) are disjoint events</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">A and \(\mathrm{B}\) are dependent events</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">A and \(\mathrm{B}\) are independent events</span> </div>

answer:
<div class="solution">\(\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{AB})\)<br/>\(\Rightarrow \frac{2}{3}=\frac{1}{3}+\mathrm{P}(\mathrm{B})-\frac{1}{6} \Rightarrow \mathrm{P}(\mathrm{B})=\frac{1}{2}\)<br/>Now, \(\mathrm{P}(\mathrm{AB})=\mathrm{P}(\mathrm{A})~ \mathrm{P}(\mathrm{B}), \mathrm{A}\) and \(\mathrm{B}\) are independent events.</div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
title:
if-mathrmptext-th-mathrmqtext-th-and-mathrmrtext-th-terms-of-hp-are-mathrmu-mathrmv-mathrmw-respectively-then-find-the-value-of-the-expression-qr-v-wr

slug:
if-mathrmptext-th-mathrmqtext-th-and-mathrmrtext-th-terms-of-hp-are-mathrmu-mathrmv-mathrmw-respectively-then-find-the-value-of-the-expression-qr-v-wr-68896

question:
<div class="question">If \(\mathrm{p}^{\text {th }}, \mathrm{q}^{\text {th }}\) and \(\mathrm{r}^{\text {th }}\) terms of H.P. are \(\mathrm{u}, \mathrm{v}, \mathrm{w}\) respectively, then find the value of the expression \((q-r) v w+(r-p) w u+(p-q) u v\).</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">Let H.P. be \(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{a}+\mathrm{d}}+\frac{1}{\mathrm{a}+2 \mathrm{~d}}+\ldots \ldots\)<br/>\(\begin{aligned}<br/>&amp; \therefore \mathrm{u}=\frac{1}{\mathrm{a}+(\mathrm{p}-1) \mathrm{d}}, \mathrm{v}=\frac{1}{\mathrm{a}+(\mathrm{q}-1) \mathrm{d}}, \\<br/>&amp; \mathrm{w}=\frac{1}{\mathrm{a}+(\mathrm{r}-1) \mathrm{d}} \Rightarrow \mathrm{a}+(\mathrm{p}-1) \mathrm{d}=\frac{1}{\mathrm{u}} \\<br/>&amp; \mathrm{a}+(\mathrm{q}-1) \mathrm{d}=\frac{1}{\mathrm{v}}, \mathrm{a}+(\mathrm{r}-1) \mathrm{d}=\frac{1}{\mathrm{w}} \\<br/>&amp; \Rightarrow(\mathrm{q}-\mathrm{r})\{\mathrm{a}+(\mathrm{p}-1) \mathrm{d}\}+(\mathrm{r}-\mathrm{p})\{\mathrm{a}+(\mathrm{q}-1) \mathrm{d}\}+\ldots \ldots \\<br/>&amp; =\frac{1}{\mathrm{u}}(\mathrm{q}-\mathrm{r})+\frac{1}{\mathrm{v}}(\mathrm{r}-\mathrm{p})+\ldots . \Rightarrow(q-r) \mathrm{vw}+\ldots . . = 0<br/>\end{aligned}\)</div>


title: if-mean-of-a-poisson-distribution-of-a-random-variable-x-is-2-then-the-value-of-px-15-is
slug: if-mean-of-a-poisson-distribution-of-a-random-variable-x-is-2-then-the-value-of-px-15-is-81006
tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
question:
<div class="question">If mean of a poisson distribution of a random variable \(X\) is 2 , then the value of \(P(X &gt; 1.5)\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{3}{e^2}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{3}{e}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(1-\frac{3}{e}\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(1-\frac{3}{e^2}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(1-\frac{3}{e^2}\)</span> </div>

answer:
<div class="solution">Since, \(\mathrm{P}(\mathrm{X}=\mathrm{r})=\frac{e^{-\lambda} \lambda^r}{r !}\) (where \(\lambda=\) mean)<br/>\(\begin{aligned}<br/>\therefore \quad &amp; \mathrm{P}(\mathrm{X}=\mathrm{r} &gt; 1.5)=\mathrm{P}(2)+\mathrm{P}(3)+\ldots \infty \\<br/>&amp; =1-\mathrm{P}(\mathrm{X}=\mathrm{r} \leq 1)=1-\mathrm{P}(0)-\mathrm{P}(1) \\<br/>&amp; =1-\left(e^{-2}+\frac{e^{-2} \times 2}{1 !}\right)=1-\frac{3}{e^2}<br/>\end{aligned}\)</div>


title: if-n-a-1000-n-b-500-and-if-n-a-b-1-and-n-a-b-p-then
slug: if-n-a-1000-n-b-500-and-if-n-a-b-1-and-n-a-b-p-then-49365
tags: ['Mathematics', 'Sets and Relations', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $n(\mathrm{~A})=1000, n(\mathrm{~B})=500$ and if $n(\mathrm{~A} \cap \mathrm{B}) \geq 1$ and $n(\mathrm{~A} \cup \mathrm{B})=p$, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$500 \leq p \leq 1000$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1001 \leq p \leq 1498$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1000 \leq p \leq 1498$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1000 \leq p \leq 1499$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1000 \leq p \leq 1499$</span> </div>

answer:
<div class="solution">$\mathrm{n}(\mathrm{A})=1000, \mathrm{n}(\mathrm{B})=500, \mathrm{n}(\mathrm{A} \cap \mathrm{B}) \geq 1$, $n(A \cup B)=p ; n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\mathrm{p}=1000+500-\mathrm{n}(\mathrm{A} \cap \mathrm{B}) \Rightarrow 1 \leq \mathrm{n}(\mathrm{A} \cap \mathrm{B}) \leq 500$<br/>Hence $p \leq 1499$ and $p \geq 1000 \Rightarrow 1000 \leq p \leq 1499$</div>


title: if-n-is-a-positive-integer-then-2-4-2-n-1-3-3-n-1-is-divisible-by
slug: if-n-is-a-positive-integer-then-2-4-2-n-1-3-3-n-1-is-divisible-by-62741
tags: ['Mathematics', 'Basic of Mathematics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $n$ is a positive integer, then $2.4^{2 n+1}+3^{3 n+1}$ is divisible by:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>7</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>11</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>27</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>11</span> </div>

answer:
<div class="solution">Let $P(n)=2 \cdot 4^{2 n+1}+3^{3 n+1}$<br/>Then $P(1)=2.4^3+3^4=209$, which is divisible by 11 but not divisible by 2,7 or 27 .<br/>Further, let $P(k)=2.4^{2 k+1}+3^{3 k+1}$ is divisible by 11 , that is,<br/>$2.4^{2 k+1}+3^{3 k+1}=11 q$ for some integer q. Now<br/>$\begin{aligned}<br/>\mathrm{P}(\mathrm{k}+1) &amp; =2 \cdot 4^{2 \mathrm{k}+3}+3^{3 \mathrm{k}+4} \\<br/>&amp; =2 \cdot 4^{2 \mathrm{k}+1} \cdot 4^2+3^{3 \mathrm{k}+1} \cdot 3^3 \\<br/>&amp; =16 \cdot 2 \cdot 4^{2 \mathrm{k}+1}+27 \cdot 3^{3 \mathrm{k}+1} \\<br/>&amp; =16 \cdot 2 \cdot 4^{2 \mathrm{k}+1}+(16+11) \cdot 3^{3 \mathrm{k}+1} \\<br/>&amp; =16\left[2 \cdot 4^{2 \mathrm{k}+1}+3^{3 \mathrm{k}+1}\right]+11 \cdot 3^{3 \mathrm{k}+1} \\<br/>&amp; =16 \cdot 11 \mathrm{q}+11 \cdot 3^{3 \mathrm{k}+1} \\<br/>&amp; =11\left(16 \mathrm{q}+3^{3 \mathrm{k}+1}\right)=11 \mathrm{~m}<br/>\end{aligned}$<br/>where $m=16 q+3^{3 k+1}$ is another integer.<br/>$\therefore \mathrm{P}(\mathrm{k}+1)$ is divisible by 11 .</div>


title: if-on-combining-two-charged-bodies-the-current-does-not-flow-then
slug: if-on-combining-two-charged-bodies-the-current-does-not-flow-then-48144
question: <div class="question">If on combining two charged bodies, the current does not flow then</div>
tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2009']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">charge is equal on both</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">capacitance is equal on both</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">potential is equal on both</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">resistance is equal on both</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">potential is equal on both</span> </div>


tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
title:
if-one-root-of-the-equation-x-2-p-x-12-0-is-4-while-the-equation-x-2-p-x-q-0-has-equal-roots-then-the-value-of-q-is-1

slug:
if-one-root-of-the-equation-x-2-p-x-12-0-is-4-while-the-equation-x-2-p-x-q-0-has-equal-roots-then-the-value-of-q-is-1-65349

question:
<div class="question">If one root of the equation $x^2+p x+12=0$ is 4, while the equation $x^2+p x+q=0$. has equal roots, then the value of ' $q$ ' is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{49}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{49}{4}$</span> </div>

answer:
<div class="solution">4 is a root of $x^2+p x+12=0$<br/>$\Rightarrow 16+4 p+12=0 \Rightarrow p=-7$<br/>Now, the equation $x^2+p x+q=0$ has equal roots.<br/>$\therefore p^2-4 q=0 \Rightarrow q=\frac{p^2}{4}=\frac{49}{4}$</div>


title: if-one-strand-of-dna-has-sequence-atcgtatg-the-sequence-in-the-complementary-strand-would-be
slug: if-one-strand-of-dna-has-sequence-atcgtatg-the-sequence-in-the-complementary-strand-would-be-31803
tags: ['Chemistry', 'Biomolecules', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">If one strand of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>DNA</mi></math> has sequence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ATCGTATG</mi></math>, the sequence in the complementary strand would be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>TAGCTTAC</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>TCACATAC</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>TAGCATAC</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>TACGATAC</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>TAGCATAC</mi></math></span> </div>

answer:
<div class="solution"><p>Due to size and geometries of bases, the only possible pairing in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>DNA</mi></math> are between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">G</mi></math> (guanine) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math> (cytosine) through three <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">H</mi></math>-bonds and between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math> (adenine) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">T</mi></math> (thymine) through two <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">H</mi></math>-bonds.</p><p>Thus, the complementary strand of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>DNA</mi></math> will be</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/f9d01d1b-61cb-4fd8-8254-611feaaced3e-image.png" style="width: 641px; height: 116px;"/></p></div>


tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
title:
if-p-1-and-p-2-be-the-length-of-perpendiculars-from-the-origin-upon-the-straight-lines-x-sec-y-cosec-a-and-x-cos-y-sin-a-cos-2-respectively-then-the-v

slug:
if-p-1-and-p-2-be-the-length-of-perpendiculars-from-the-origin-upon-the-straight-lines-x-sec-y-cosec-a-and-x-cos-y-sin-a-cos-2-respectively-then-the-v-63723

question:
<div class="question">If $P_{1}$ and $P_{2}$ be the length of perpendiculars from the origin upon the straight lines $x \sec \theta+y \operatorname{cosec} \theta$ $=a$ and $x \cos \theta-y \sin \theta=a \cos 2 \theta$ respectively,
<br/> <br/>then the value of $4 \mathrm{P}_{1}^{2}+\mathrm{P}_{2}^{2}$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$a^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 a^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a^{2} / 2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 a^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a^{2}$</span> </div>

answer:
<div class="solution">We have $P_{1}=$ length of perpendicular from (0,0) on $x \sec \theta+y \operatorname{cosec} \theta=a$
<br/> <br/>$\begin{aligned} \text { i.e. } P_{1} &amp;=\left|\frac{a}{\sqrt{\sec ^{2} \theta+\cos e c^{2} \theta}}\right|=|a \sin \theta \cos \theta| \\ &amp;=\left|\frac{a}{2} \sin 2 \theta\right| \text { or } 2 P_{1}=|a \sin 2 \theta| \end{aligned}$
<br/> <br/>$\mathrm{P}_{2}=$ Length of the perpendicular from (0,0) on $x \cos \theta-y \sin \theta=a \cos 2 \theta$
<br/> <br/>$P_{2}=\left|\frac{a \cos 2 \theta}{\sqrt{\cos ^{2} \theta+\sin ^{2} \theta}}\right|=|a \cos 2 \theta|$
<br/> <br/>Now, $4 P_{1}^{2}+P_{2}^{2}=a^{2} \sin ^{2} 2 \theta+a^{2} \cos ^{2} 2 \theta=a^{2}$</div>


tags: ['Mathematics', 'Mathematical Reasoning', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
if-p-4-is-an-even-prime-number-q-6-is-a-divisor-of-12-and-r-the-hcf-of-4-and-6-is-2-then-which-of-the-following-is-correct

slug:
if-p-4-is-an-even-prime-number-q-6-is-a-divisor-of-12-and-r-the-hcf-of-4-and-6-is-2-then-which-of-the-following-is-correct-73005

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>:</mo><mn>4</mn></math> is an even prime number, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>:</mo><mn>6</mn></math> is a divisor of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>:</mo></math> the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>HCF</mi></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>, then which of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>p</mi><mo>∧</mo><mi>q</mi></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>p</mi><mo>∨</mo><mi>q</mi></mrow></mfenced><mo>∧</mo><mo>~</mo><mi>r</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>~</mo><mfenced><mrow><mi>q</mi><mo>∧</mo><mi>r</mi></mrow></mfenced><mo>∨</mo><mi>p</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>~</mo><mi>p</mi><mo>∨</mo><mfenced><mrow><mi>q</mi><mo>∧</mo><mi>r</mi></mrow></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>~</mo><mi>p</mi><mo>∨</mo><mfenced><mrow><mi>q</mi><mo>∧</mo><mi>r</mi></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>:</mo><mn>4</mn></math> is an even prime number,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>:</mo><mn>6</mn></math> is a divisor of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> and</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>:</mo></math> the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>HCF</mi></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>.</p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>~</mo><mi>p</mi><mo>∨</mo><mfenced><mrow><mi>q</mi><mo>∧</mo><mi>r</mi></mrow></mfenced></math> is correct.</p><p>Hence, option (d) is correct.</p></div>


title: if-p-a-b-c-b-c-a-c-a-b-a-b-c-1-p-t-p-i-then-the-value-of-a-3-b-3-c-3-is
slug: if-p-a-b-c-b-c-a-c-a-b-a-b-c-1-p-t-p-i-then-the-value-of-a-3-b-3-c-3-is-49037
tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfenced close="]" open="["><mtable columnalign="left"><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mi>c</mi><mo>=</mo><mn>1</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>P</mi><mi>T</mi></msup><mi>P</mi><mo>=</mo><mi>I</mi></math> then the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>3</mn></msup><mo>+</mo><msup><mi>b</mi><mn>3</mn></msup><mo>+</mo><msup><mi>c</mi><mn>3</mn></msup></math> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </div>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfenced close="]" open="["><mtable columnalign="left"><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>P</mi><mi>T</mi></msup><mo>=</mo><mfenced close="]" open="["><mtable columnalign="left"><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math></p><p>Also given <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>P</mi><mi>T</mi></msup><mi>P</mi><mo>=</mo><mi>I</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfenced close="|" open="|"><mrow><msup><mi>P</mi><mi>T</mi></msup><mi>P</mi></mrow></mfenced><mo>=</mo><mfenced close="|" open="|"><mi>I</mi></mfenced><mo>=</mo><mn>1</mn></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced close="|" open="|"><mtable columnalign="left"><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>⇒</mo><mfenced close="|" open="|"><mtable columnalign="left"><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mo>±</mo><mn>1</mn></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>a</mi><mi>b</mi><mi>c</mi><mo>-</mo><msup><mi>a</mi><mn>3</mn></msup><mo>-</mo><msup><mi>b</mi><mn>3</mn></msup><mo>-</mo><msup><mi>c</mi><mn>3</mn></msup><mo>=</mo><mo>±</mo><mn>1</mn></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>∓</mo><mn>1</mn><mo>=</mo><msup><mi>a</mi><mn>3</mn></msup><mo>+</mo><msup><mi>b</mi><mn>3</mn></msup><mo>+</mo><msup><mi>c</mi><mn>3</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>a</mi><mn>3</mn></msup><mo>+</mo><msup><mi>b</mi><mn>3</mn></msup><mo>+</mo><msup><mi>c</mi><mn>3</mn></msup><mo>=</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></p></div>


tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
if-p-a-q-b-r-c-and-the-system-of-equations-p-x-a-y-a-z-0-b-x-q-y-b-z-0-c-x-cy-rz-0-has-a-nontrivial-solution-then-the-value-of-p-a-p-q-b-q-r-c-r-is

slug:
if-p-a-q-b-r-c-and-the-system-of-equations-p-x-a-y-a-z-0-b-x-q-y-b-z-0-c-x-cy-rz-0-has-a-nontrivial-solution-then-the-value-of-p-a-p-q-b-q-r-c-r-is-87009

question:
<div class="question">If $p \neq a, q \neq b, r \neq c$ and the system of equations
<br/>$\begin{aligned}
<br/>&amp; p x+a y+a z=0 \\
<br/>&amp; b x+q y+b z=0 \\
<br/>&amp; c x+c y+r z=0
<br/>\end{aligned}$
<br/>has a non-trivial solution, then the value of 
<br/>$\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$1$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$2$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$\frac{1}{2}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$2$
<br/></span> </div>

answer:
<div class="solution">As the given system of equations has a non-trivial solution.
<br/>$\Delta=\left|\begin{array}{lll}p &amp; a &amp; a \\ b &amp; q &amp; b \\ c &amp; c &amp; r\end{array}\right|=0$
<br/>Applying $C_2 \rightarrow C_2-C_1$ and $C_3 \rightarrow C_3-C_1$
<br/>$\Delta=\left|\begin{array}{ccc}p &amp; a-p &amp; a-p \\ b &amp; q-b &amp; 0 \\ c &amp; 0 &amp; r-c\end{array}\right|=0$
<br/>Expanding along $C_3$, we get
<br/>$(a-p)\left|\begin{array}{cc}b &amp; q-b \\ c &amp; 0\end{array}\right|+(r-c)\left|\begin{array}{cc}p &amp; a-p \\ b &amp; q-b\end{array}\right|=0$
<br/>$\begin{aligned}
<br/>&amp; \Rightarrow(a-p)(-c)(q-b)+(r-c) \\
<br/>&amp; \{p(q-b)-b(a-p)\}=0 \\
<br/>&amp; \Rightarrow \\
<br/>&amp; (p-a)(q-b) c+p(r-c)(q-b)+b(r-c)(p-a)=0
<br/>\end{aligned}$
<br/>Dividing by $(p-a)(q-b)(r-c)$, we get
<br/>$\frac{c}{r-c}+\frac{p}{p-a}+\frac{b}{q-b}=0$
<br/>$\Rightarrow \frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}=\frac{q-b}{q-b}+\frac{r-c}{r-c}=2$</div>


tags: ['Mathematics', 'Ellipse', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
title:
if-p-and-p-denote-the-lengths-of-the-perpendicular-from-a-focus-and-the-centre-of-an-ellipse-with-semimajor-axis-of-length-a-respectively-on-a-tangent

slug:
if-p-and-p-denote-the-lengths-of-the-perpendicular-from-a-focus-and-the-centre-of-an-ellipse-with-semimajor-axis-of-length-a-respectively-on-a-tangent-52588

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mo>'</mo></msup></math> denote the lengths of the perpendicular from a focus and the centre of an ellipse with semi-major axis of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>, respectively, on a tangent to the ellipse and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> denotes the focal distance of the point, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>p</mi><mo>=</mo><mi>r</mi><msup><mi>p</mi><mo>'</mo></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mi>p</mi><mo>=</mo><mi>a</mi><msup><mi>p</mi><mo>'</mo></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>p</mi><mo>=</mo><mi>r</mi><msup><mi>p</mi><mo>'</mo></msup><mo>+</mo><mn>1</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><msup><mi>p</mi><mo>'</mo></msup><mo>+</mo><mi>r</mi><mi>p</mi><mo>=</mo><mn>1</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>p</mi><mo>=</mo><mi>r</mi><msup><mi>p</mi><mo>'</mo></msup></math></span> </div>

answer:
<div class="solution"><p>Let the ellipse be <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>x</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn></math></p><p>and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mi>cos</mi><mi>θ</mi></mrow><mi>a</mi></mfrac><mo>+</mo><mfrac><mrow><mi>y</mi><mi>sin</mi><mi>θ</mi></mrow><mi>b</mi></mfrac><mo>=</mo><mn>1</mn></math>  <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math></p><p>be a tangent to it at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mi>cos</mi><mi>θ</mi><mo>,</mo><mi>b</mi><mi>sin</mi><mi>θ</mi><mo>)</mo></math>. Then, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo></math> length of the perpendicular from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>(</mo><mi>a</mi><mi>e</mi><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> on Eq. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfenced close="|" open="|"><mfrac><mrow><mi>e</mi><mi>cos</mi><mi>θ</mi><mo>-</mo><mn>1</mn></mrow><msqrt><mfrac><mrow><msup><mi>cos</mi><mn>2</mn></msup><mi>θ</mi></mrow><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><msup><mi>sin</mi><mn>2</mn></msup><mi>θ</mi></mrow><msup><mi>b</mi><mn>2</mn></msup></mfrac></msqrt></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mo>'</mo></msup><mo>=</mo></math> length of the perpendicular from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math> on Eq. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>p</mi><mo>'</mo></msup><mo>=</mo><mfenced close="|" open="|"><mfrac><mn>1</mn><msqrt><mfrac><mrow><msup><mi>cos</mi><mn>2</mn></msup><mi>θ</mi></mrow><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><msup><mi>sin</mi><mn>2</mn></msup><mi>θ</mi></mrow><msup><mi>b</mi><mn>2</mn></msup></mfrac></msqrt></mfrac></mfenced></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>a</mi><mi>e</mi><mi>cos</mi><mi>θ</mi><mo>-</mo><mi>a</mi></math></p><p>Clearly, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><msup><mi>p</mi><mo>'</mo></msup><mo>=</mo><mi>a</mi><mi>p</mi></math></p></div>


title: if-p-b-5-3-p-a-b-2-1-and-p-a-b-5-4-then-p-a-b-p-a-b
slug: if-p-b-5-3-p-a-b-2-1-and-p-a-b-5-4-then-p-a-b-p-a-b-59374
tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{P}(\mathrm{B})=\frac{3}{5}, \mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{1}{2}$ and $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{4}{5}$, then $\mathrm{P}(\mathrm{A} \cup \mathrm{B})^{\prime}+\mathrm{P}\left(\mathrm{A}^{\prime} \cup \mathrm{B}\right)=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{5}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{4}{5}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$P(B)=\frac{3}{5}, P(A \mid B)=\frac{1}{2}$ and $P(A \cup B)=\frac{4}{5}$<br/>$P(A \cap B)=P(A \mid B) P(B)=\frac{1}{2} \cdot \frac{3}{5}=\frac{3}{10}$<br/>$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$\mathrm{P}(\mathrm{A})=\frac{4}{5}-\frac{3}{10}=\frac{1}{2} . \quad \mathrm{P}\left(\mathrm{A}^{\prime}\right)=1-\mathrm{P}(\mathrm{A})=\frac{1}{2}$<br/>We know, $\mathrm{P}(\mathrm{A} \cap \mathrm{B})+\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)=\mathrm{P}(\mathrm{B})$<br/>[as $A \cap B$ and $A^{\prime} \cap B$ are mutually exclusive events]<br/>$\Rightarrow \quad \frac{3}{10}+\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)=\frac{3}{5}$<br/>$\Rightarrow \mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)=\frac{3}{5}-\frac{3}{10}=\frac{3}{10}$<br/>Now, $\mathrm{P}\left(\mathrm{A}^{\prime} \cup \mathrm{B}\right)=\mathrm{P}\left(\mathrm{A}^{\prime}\right)+\mathrm{P}(\mathrm{B})-\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)$<br/>$=\frac{1}{2}+\frac{3}{5}-\frac{3}{10}=\frac{5+6-3}{10}=\frac{4}{5}$<br/>$\mathrm{P}\left((\mathrm{A} \cup \mathrm{B})^{\prime}\right)=1-\mathrm{P}(\mathrm{A} \cup \mathrm{B})=1-\frac{4}{5}=\frac{1}{5}$<br/>$\therefore \mathrm{P}\left((\mathrm{A} \cup \mathrm{B})^{\prime}\right)+\mathrm{P}\left(\mathrm{A}^{\prime} \cup \mathrm{B}\right)=\frac{1}{5}+\frac{4}{5}=1$</div>


title: if-p-p-x-p-x-q-y-q-q-y-r-z-r-z-r-0-then-the-value-of-x-p-y-q-z-r-is
slug: if-p-p-x-p-x-q-y-q-q-y-r-z-r-z-r-0-then-the-value-of-x-p-y-q-z-r-is-85490
tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left|\begin{array}{ccc}\mathrm{p} &amp; \mathrm{q}-\mathrm{y} &amp; \mathrm{r}-\mathrm{z} \\ \mathrm{p}-\mathrm{x} &amp; \mathrm{q} &amp; \mathrm{r}-\mathrm{z} \\ \mathrm{p}-\mathrm{x} &amp; \mathrm{q}-\mathrm{y} &amp; \mathrm{r}\end{array}\right|=0,$ then the value of
<br/> <br/>$\frac{p}{x}+\frac{q}{y}+\frac{r}{z}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 \mathrm{pqr}$</span> </li> </ul>

answer:
<div class="solution">$\left|\begin{array}{ccc}\mathrm{p} &amp; \mathrm{q}-\mathrm{y} &amp; \mathrm{r}-\mathrm{z} \\ \mathrm{p}-\mathrm{x} &amp; \mathrm{q} &amp; \mathrm{r}-\mathrm{z} \\ \mathrm{p}-\mathrm{x} &amp; \mathrm{q}-\mathrm{y} &amp; \mathrm{r}\end{array}\right|=0$
<br/> <br/>Apply $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{3}$ and $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3},$ we get
<br/> <br/>$\left|\begin{array}{ccc}\mathrm{x} &amp; 0 &amp; -\mathrm{z} \\ 0 &amp; \mathrm{y} &amp; -\mathrm{z} \\ \mathrm{p}-\mathrm{x} &amp; \mathrm{q}-\mathrm{y} &amp; \mathrm{r}\end{array}\right|=0$
<br/> <br/>$\Rightarrow \mathrm{x}[\mathrm{yr}+\mathrm{z}(\mathrm{q}-\mathrm{y})]-\mathrm{z}[0-\mathrm{y}(\mathrm{p}-\mathrm{x})]=0$
<br/> <br/>[Expansion along first row $]$
<br/> <br/>$\Rightarrow x y r+z x q+y z p=2 x y z \Rightarrow \frac{p}{x}+\frac{q}{y}+\frac{r}{z}=2$</div>


title: if-p-raju-is-tall-and-q-raju-is-intelligent-then-the-symbolic-statement-sim-p-vee-q-means
slug: if-p-raju-is-tall-and-q-raju-is-intelligent-then-the-symbolic-statement-sim-p-vee-q-means-65575
tags: ['Mathematics', 'Statistics', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
question:
<div class="question">If: \(p\) Raju is tall and \(q\) : Raju is intelligent, then the symbolic statement \(\sim p \vee q\) means</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Raju is not tall or he is intelligent.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Raju is tall or he is intelligent</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Raju is not tall and he is intelligent</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Raju is not tall implies he is intelligent</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Raju is not tall or he is intelligent.</span> </div>

answer:
<div class="solution">\(\sim \mathrm{p} \vee \mathrm{q}:\) Raju is not tall or he is intelligent.</div>


tags: ['Physics', 'Units and Dimensions', 'BITSAT', 'BITSAT 2010']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P1.db
title:
if-p-represents-radiation-pressure-c-represents-speed-of-light-and-q-represents-radiation-energy-striking-a-unit-area-per-second-the-nonzero-integers-

slug:
if-p-represents-radiation-pressure-c-represents-speed-of-light-and-q-represents-radiation-energy-striking-a-unit-area-per-second-the-nonzero-integers-26958

question:
<div class="question">If $P$ represents radiation pressure, c represents speed of light and $\mathrm{Q}$ represents radiation energy striking a unit area per second, the non-zero integers $x, y$ and $z$ such that $P^{x} Q^{y} c^{z}$ is dimensionless, are.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x=1, y=1, z=-1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x=1, y=-1, z=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad x=-1, y=1, z=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x=1, y=1, z=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x=1, y=-1, z=1$</span> </div>


title: if-r-0-n-r-1-r-2-n-c-r-6-2-8-1-then-n
slug: if-r-0-n-r-1-r-2-n-c-r-6-2-8-1-then-n-79887
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sum_{r=0}^{n} \frac{r+2}{r+1}^{n} C_{r}=\frac{2^{8}-1}{6},$ then $n=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">6</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\sum_{r=0}^{n} \frac{r+2}{r+1}{ }^{n} C_{r}=\frac{2^{8}-1}{6}$
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>&amp; \Rightarrow \sum_{r=0}^{n}\left[1+\frac{1}{r+1}\right]{ }^{n} C_{r}=\frac{2^{8}-1}{6} \\
<br/> <br/>&amp; \Rightarrow 2^{n}+\sum_{r=0}^{n} \frac{1}{n+1} \cdot{ }^{n+1} C_{r+1}=\frac{2^{8}-1}{6} \\
<br/> <br/>\Rightarrow 2^{n}+&amp; \frac{2^{n+1}-1}{n+1}=\frac{2^{8}-1}{6} \Rightarrow \frac{2^{n}(n+3)-1}{n+1}=\frac{2^{8}-1}{6} \\
<br/> <br/>\Rightarrow &amp; \frac{2^{n}(n+1+2)-1}{n+1}=\frac{2^{5}(6+2)-1}{6}
<br/> <br/>\end{aligned}
<br/> <br/>$$
<br/> <br/>Comparing we get $\mathrm{n}+1=6 \Rightarrow \mathrm{n}=5$</div>


title: if-r-1-n-t-r-n-n-1-n-2-n-3-8-where-t-r-denotes-the-r-t-h-term-of-a-series-then-lim-n-r-1-n-1-t-r-is
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
slug:
if-r-1-n-t-r-n-n-1-n-2-n-3-8-where-t-r-denotes-the-r-t-h-term-of-a-series-then-lim-n-r-1-n-1-t-r-is-81041

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>t</mi><mi>r</mi></msub><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfenced></mrow><mn>8</mn></mfrac><mo>,</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mi>r</mi></msub></math> denotes the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mrow><mi>t</mi><mi>h</mi></mrow></msup></math> term of a series, then <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mn>1</mn><msub><mi>t</mi><mi>r</mi></msub></mfrac></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>8</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </li> </ul>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>t</mi><mi>r</mi></msub><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow></mfenced></mrow><mn>8</mn></mfrac><mo>=</mo><msub><mi>S</mi><mi>n</mi></msub></math> (say)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></munderover><msub><mi>t</mi><mi>r</mi></msub><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><mn>8</mn></mfrac><mo>=</mo><msub><mi>S</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></math></p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mi>n</mi></msub><mo>=</mo><msub><mi>S</mi><mi>n</mi></msub><mo>-</mo><msub><mi>S</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mn>1</mn><msub><mi>t</mi><mi>r</mi></msub></mfrac><mo>=</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mn>2</mn><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfenced><mrow><mfrac><mn>1</mn><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfenced><mrow><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfenced><mrow><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfenced><mrow><mn>0</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p></div>


tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
title:
if-r-and-s-arenonzero-constant-vectors-and-the-scalar-b-is-chosen-such-that-r-b-s-is-minimum-then-the-value-of-b-s-2-r-b-s-2-is-equal-to

slug:
if-r-and-s-arenonzero-constant-vectors-and-the-scalar-b-is-chosen-such-that-r-b-s-is-minimum-then-the-value-of-b-s-2-r-b-s-2-is-equal-to-78020

question:
<div class="question">If $\vec{r}$ and $\vec{s}$ arenon-zero constant vectors and the scalar $b$ is chosen such that $|\vec{r}+b \vec{s}|$ is minimum, then the value of $|b \vec{s}|^{2}+|\vec{r}+b \vec{s}|^{2}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2|\vec{r}|^{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$|\vec{r}|^{2} / 2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3|\vec{r}|^{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$|\vec{r}|^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$|\vec{r}|^{2}$</span> </div>

answer:
<div class="solution">For minimum value $|\vec{r}+b \vec{s}|=0$. Let $\vec{r}$ and $\vec{s}$ are anti-parallel so $b \vec{s}=-\vec{r}$
<br/> <br/>$\therefore|b \vec{s}|^{2}+|\vec{r}+b \vec{s}|^{2}=|-\vec{r}|^{2}+|\vec{r}-\vec{r}|^{2}=|\vec{r}|^{2}$</div>


tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
title:
if-r-land-xafracarfracar2ldots-to-infty-mathrmymathrmbfracmathrmbmathrmrfracmathrmbmathrmr2ldots-text-to-infty-and-zcfraccr2fraccr4ldots-to-infty-then

slug:
if-r-land-xafracarfracar2ldots-to-infty-mathrmymathrmbfracmathrmbmathrmrfracmathrmbmathrmr2ldots-text-to-infty-and-zcfraccr2fraccr4ldots-to-infty-then-51687

question:
<div class="question">If \(|r| &gt; \) land \(x=a+\frac{a}{r}+\frac{a}{r^2}+\ldots\) to \(\infty\),<br/>\(\mathrm{y}=\mathrm{b}-\frac{\mathrm{b}}{\mathrm{r}}+\frac{\mathrm{b}}{\mathrm{r}^2}-\ldots . \text { to } \infty\)<br/>and \(z=c+\frac{c}{r^2}+\frac{c}{r^4}+\ldots\) to \(\infty\), then \(\frac{x y}{z}=\)</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\frac{\mathrm{ab}}{\mathrm{c}}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{\mathrm{ac}}{\mathrm{b}}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{\mathrm{bc}}{\mathrm{a}}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{\mathrm{ab}}{\mathrm{c}}\)</span> </div>

answer:
<div class="solution">Since \(|\mathrm{r}| &gt; 1, \frac{1}{|\mathrm{r}|}  &lt; 1\)<br/>\(\therefore \mathrm{x}=\frac{\mathrm{a}}{1-\frac{1}{\mathrm{r}}}=\frac{\mathrm{ar}}{\mathrm{r}-1}\)<br/>Similarly, \(y=\frac{b}{1-\left(-\frac{1}{r}\right)}=\frac{b r}{r+1}\) and<br/>\(\begin{gathered}<br/>z=\frac{c}{1-\frac{1}{r^2}}=\frac{cr^2}{r^2-1} \quad ...(1)\\<br/>\therefore \quad x y=\frac{a r}{r-1} \times \frac{b r}{r+1}=\frac{a b r^2}{r^2-1} \quad ...(2)<br/>\end{gathered}\)<br/>Dividing (2) by (1), we get<br/>\(\frac{\mathrm{xy}}{\mathrm{z}}=\frac{\mathrm{abr}^2}{\mathrm{r}^2-1} \times \frac{\mathrm{r}^2-1}{\mathrm{cr}^2}=\frac{\mathrm{ab}}{\mathrm{c}}\)</div>


title: if-s-i-n-x-s-i-n-x-dx-a-x-b-lo-g-sin-x-c-then-value-of-a-b-is
slug: if-s-i-n-x-s-i-n-x-dx-a-x-b-lo-g-sin-x-c-then-value-of-a-b-is-31859
tags: ['Mathematics', 'Indefinite Integration', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\int \frac{\sin x}{\sin (x-\alpha)} \mathrm{d} \mathrm{x}=\mathrm{A} x+\mathrm{B} \log \sin (x-\alpha)+\mathrm{C}$, then value of $(A, B)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(-\cos \alpha, \sin \alpha)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(\cos \alpha, \sin \alpha)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(-\sin \alpha, \cos \alpha)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(\sin \alpha, \cos \alpha)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(\cos \alpha, \sin \alpha)$</span> </div>

answer:
<div class="solution">$\int \frac{\sin x}{\sin (x-\alpha)} \mathrm{dx}=\int \frac{\sin (x-\alpha+\alpha}{\sin (x-\alpha)} \mathrm{dx}$
<br/> <br/>$=\int \frac{\sin (x-\alpha) \cos \alpha+\cos (x-\alpha) \sin \alpha}{\sin (x-\alpha)} \mathrm{d} \mathrm{x}$
<br/> <br/>$=\int\{\cos \alpha+\sin \alpha \cot (x-\alpha)\} \mathrm{d} \mathrm{x}$
<br/> <br/>$=(\cos \alpha) x+(\sin \alpha) \log \sin (x-\alpha)+\mathrm{C}$
<br/> <br/>$\therefore \mathrm{A}=\cos \alpha, \mathrm{B}=\sin \alpha$</div>


tags: ['Mathematics', 'Statistics', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
if-sample-a-contains-100-observations-101-102-200-and-sample-b-contains-100-obsections-151-152-ldots-250-then-ratio-of-variance-v-a-v-b

slug:
if-sample-a-contains-100-observations-101-102-200-and-sample-b-contains-100-obsections-151-152-ldots-250-then-ratio-of-variance-v-a-v-b-18905

question:
<div class="question">If sample A contains 100 observations $101,102, \ldots$ 200 and sample B contains 100 obsections $151,152,$ \ldots 250, then ratio of variance $\mathrm{v}_{\mathrm{A}} / \mathrm{v}_{\mathrm{B}}=$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{9}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{4}{9}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{3}$</span> </li> </ul>

answer:
<div class="solution">$\because \quad \delta_{\mathrm{x}}^{2}=\frac{\Sigma \mathrm{d}^{2} \mathrm{i}}{\mathrm{n}}$
<br/>But both $A$ and $B$ have 100 observations, then both the sample $A$ and $B$ have same standard deviation and the same variance.
<br/>Hence, $\frac{\mathrm{V}_{\mathrm{A}}}{\mathrm{V}_{\mathrm{B}}}=1$</div>


title: if-sec-2-3-4-then-the-general-value-of-is
slug: if-sec-2-3-4-then-the-general-value-of-is-42496
question: <div class="question">If $\sec ^2 \theta=\frac{4}{3}$, then the general value of $\theta$ is</div>
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{n} \pi \pm \frac{\pi}{6}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{n} \pi \pm \frac{\pi}{6}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \mathrm{n} \pi \pm \frac{\pi}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{n} \pi \pm \frac{\pi}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{n} \pi \pm \frac{\pi}{6}$</span> </div>

answer:
<div class="solution">We have $\sec ^2 \theta=\frac{4}{3} \Rightarrow \cos ^2 \theta=\frac{3}{4}$<br/>$\Rightarrow \cos ^2 \theta=\cos ^2\left(\frac{\pi}{6}\right) \Rightarrow$<br/>$\theta=n \pi \pm \frac{\pi}{6} \ldots \ldots \ldots$</div>


title: if-sin-1-1-a-2-2-a-cos-1-1-b-2-1-b-2-tan-1-1-x-2-2-x-then-what-is-the-value-of-x-1
slug: if-sin-1-1-a-2-2-a-cos-1-1-b-2-1-b-2-tan-1-1-x-2-2-x-then-what-is-the-value-of-x-1-13124
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1}\left(\frac{2 a}{1+a^{2}}\right)-\cos ^{-1}\left(\frac{1-b^{2}}{1+b^{2}}\right)=\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)$,
<br/> <br/>then what is the value of $x$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a / b$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a b$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$b / a$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{a-b}{1+a b}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{a-b}{1+a b}$</span> </div>

answer:
<div class="solution">Given,
<br/> <br/>$\sin ^{-1}\left(\frac{2 a}{1+a^{2}}\right)-\cos ^{-1}\left(\frac{1-b^{2}}{1+b^{2}}\right)=\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)$
<br/> <br/>$\therefore \quad 2 \tan ^{-1} a-2 \tan ^{-1} b=2 \tan ^{-1} x$
<br/> <br/>$\Rightarrow \tan ^{-1} a-\tan ^{-1} b=\tan ^{-1} x$
<br/> <br/>$\Rightarrow \quad \tan ^{-1}\left(\frac{a-b}{1+a b}\right)=\tan ^{-1} x$
<br/> <br/>$\Rightarrow \quad x=\frac{a-b}{1+a b}$</div>


title: if-sin-1-sin-2-sin-3-3-then-cos-1-cos-2-cos-3
slug: if-sin-1-sin-2-sin-3-3-then-cos-1-cos-2-cos-3-86027
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \theta_{1}+\sin \theta_{2}+\sin \theta_{3}=3,$ then $\cos \theta_{1}+\cos \theta_{2}$$+\cos \theta^{3}=$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">Since, $\sin \theta_{1}+\sin \theta_{2}+\sin \theta_{3}=3$
<br/> <br/>$\begin{array}{ll}\therefore &amp; \sin \theta_{1}=\sin \theta_{2}=\sin \theta_{3}=1 \Rightarrow \theta_{1}=\theta_{2}=\theta_{3}= \\ &amp; \frac{\pi}{2} \\ \therefore &amp; \cos \theta_{1}+\cos \theta_{2}+\cos \theta_{3}=0\end{array}$</div>


tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
title:
if-sin-1-x-sin-1-y-sin-1-z-3-2-and-f-1-2-f-p-q-f-p-f-q-p-q-r-then-x-f-1-y-f-2-z-f-3-x-y-z-x-f-1-y-f-2-z-f-3-is-equal-to

slug:
if-sin-1-x-sin-1-y-sin-1-z-3-2-and-f-1-2-f-p-q-f-p-f-q-p-q-r-then-x-f-1-y-f-2-z-f-3-x-y-z-x-f-1-y-f-2-z-f-3-is-equal-to-57523

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>+</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>y</mi><mo>+</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>z</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac></math> and<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>2</mn><mo>,</mo><mi>f</mi><mo>(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>p</mi><mo>)</mo><mo>·</mo><mi>f</mi><mo>(</mo><mi>q</mi><mo>)</mo><mo>,</mo><mo>∀</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mi>R</mi></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mrow><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>+</mo><msup><mi>y</mi><mrow><mi>f</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow><mi>f</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msup><mo>-</mo><mfrac><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>)</mo></mrow><mrow><msup><mi>x</mi><mrow><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>+</mo><msup><mi>y</mi><mrow><mi>f</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow><mi>f</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msup></mrow></mfrac></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo><mo> </mo><mfrac><mrow><mo>-</mo><mi>π</mi></mrow><mn>2</mn></mfrac><mo>≤</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>≤</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>,</mo><mfrac><mrow><mo>-</mo><mi>π</mi></mrow><mn>2</mn></mfrac><mo>≤</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>y</mi><mo>≤</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mi>π</mi></mrow><mn>2</mn></mfrac><mo>≤</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>z</mi><mo>≤</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math></p><p>Given that, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>+</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>y</mi><mo>+</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>z</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac></math></p><p>Which is possible only when</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>y</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>z</mi><mo>=</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>y</mi><mo>=</mo><mi>z</mi><mo>=</mo><mn>1</mn></math></p><p>Put <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mi>q</mi><mo>=</mo><mn>1</mn></math></p><p>Then, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>2</mn><mo>×</mo><mn>2</mn><mo>=</mo><mn>4</mn></math></p><p>and put <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>=</mo><mn>2</mn></math></p><p>then, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>3</mn><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo> </mo><mi>f</mi><mo>(</mo><mn>2</mn><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>·</mo><msup><mn>2</mn><mn>2</mn></msup><mo>=</mo><mn>8</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>x</mi><mrow><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>+</mo><msup><mi>y</mi><mrow><mi>f</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow><mi>f</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msup><mo>-</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi></mrow><mrow><msup><mi>x</mi><mrow><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>+</mo><msup><mi>y</mi><mrow><mi>f</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow><mi>f</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msup></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>-</mo><mfrac><mn>3</mn><mrow><mn>1</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>3</mn><mo>-</mo><mn>1</mn><mo>=</mo><mn>2</mn></math></p></div>


title: if-sin-1-x-tan-1-x-2-then-2-x-2-1
slug: if-sin-1-x-tan-1-x-2-then-2-x-2-1-72010
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>+</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>=</mo><mi>π</mi><mo>/</mo><mn>2</mn></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>=</mo></math></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>5</mn></msqrt></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msqrt><mn>5</mn></msqrt><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msqrt><mn>5</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>5</mn></msqrt></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>+</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>=</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>⇒</mo><mn>0</mn><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>=</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>-</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>=</mo><msup><mi>cos</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt><mi>x</mi></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn><mo>⇒</mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn><mo>±</mo><msqrt><mn>5</mn></msqrt></mrow><mn>2</mn></mfrac></math></p><p>as <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>&gt;</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><msqrt><mn>5</mn></msqrt><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>=</mo><msqrt><mn>5</mn></msqrt></math></p></div>


title: if-sin-2-sin-2-12-cos-2-cos-2-32-then-cos-2-is-equal-to
slug: if-sin-2-sin-2-12-cos-2-cos-2-32-then-cos-2-is-equal-to-81474
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{2} \theta+\sin ^{2} \phi=1 / 2, \cos ^{2} \theta+\cos ^{2} \phi=3 / 2,$ then $\cos ^{2}(\theta-\phi)$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 / 8$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$5 / 8$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 / 4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5 / 4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5 / 8$</span> </div>

answer:
<div class="solution">Using cosine formula 
<br/>$2 \sin (\theta+\phi) \cos (\theta-\phi)=1 / 2$ ...(i)
<br/>$2 \cos (\theta+\phi) \cos (\theta-\phi)=3 / 2$ ...(ii)
<br/>Squaring (1) and (2) and then adding $\cos ^{2}(\theta-\phi)=\frac{1}{4}+\frac{9}{4}=\frac{5}{2} \Rightarrow \cos ^{2}(\theta-\phi)=\frac{5}{8}$</div>


title: if-sin-thetafrac12-and-tan-theta1-sqrt3-then-theta
slug: if-sin-thetafrac12-and-tan-theta1-sqrt3-then-theta-49925
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
question:
<div class="question">If \(\sin \theta=-\frac{1}{2}\) and \(\tan \theta=1 / \sqrt{3}\), then \(\theta=\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(2 \mathrm{n} \pi+\pi / 6\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(2 \mathrm{n} \pi+11 \pi / 6\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(2 \mathrm{n} \pi+7 \pi / 6\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(2 \mathrm{n} \pi+\pi / 4\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(2 \mathrm{n} \pi+7 \pi / 6\)</span> </div>

answer:
<div class="solution">We shall first consider values of \(\theta\) between 0<br/>\(\begin{aligned}<br/>&amp; \text { and } 2 \pi \sin \theta=-\frac{1}{2}=-\sin \frac{\pi}{6}=\sin \left(\pi+\frac{\pi}{6}\right) \\<br/>&amp; \text { or } \sin (2 \pi-\pi / 6) \\<br/>&amp; \therefore \theta=7 \pi / 6 ; 11 \pi / 6 \\<br/>&amp; \tan \theta=1 / \sqrt{3}=\tan (\pi / 6) \text { or } \tan (\pi+\pi / 6) \\<br/>&amp; \therefore \theta=\pi / 6,7 \pi / 6<br/>\end{aligned}\)<br/>The value of \(\theta\) which satisfies both the equations is \(7 \pi / 6\)<br/>Hence the general value of \(\theta\) is \(2 n \pi+7 \pi / 6\) where \(\mathrm{n} \in \mathrm{I}\)</div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
if-sound-travels-in-air-with-the-speed-of-340-m-s-1-then-the-number-of-tones-present-in-an-open-organ-pipe-of-length-2-m-for-a-maximum-frequency-of-12

slug:
if-sound-travels-in-air-with-the-speed-of-340-m-s-1-then-the-number-of-tones-present-in-an-open-organ-pipe-of-length-2-m-for-a-maximum-frequency-of-12-44624

question:
<div class="question">If sound travels in air with the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>340</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, then the number of tones present in an open organ pipe of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">m</mi></math> for a maximum frequency of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1200</mn><mo> </mo><mi>Hz</mi></math>, are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>17</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn></math></span> </div>

answer:
<div class="solution"><p>Given, speed of sound, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>340</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math><br/>Length of open organ pipe, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>=</mo><mn>2</mn><mo> </mo><mi mathvariant="normal">m</mi></math></p><p>Frequency, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mn>1200</mn><mo> </mo><mi>Hz</mi></math><br/>For open organ pipe,</p><p>Fundamental frequency, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub><mo>=</mo><mfrac><mi>v</mi><mrow><mn>2</mn><mi>l</mi></mrow></mfrac><mo>=</mo><mfrac><mn>340</mn><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mn>85</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Number of tones present in the open organ pipe<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi>f</mi><msub><mi>f</mi><mn>0</mn></msub></mfrac><mo>=</mo><mfrac><mn>1200</mn><mn>85</mn></mfrac><mo>=</mo><mn>14</mn><mo>.</mo><mn>11</mn><mo>≃</mo><mn>14</mn></math></p></div>


tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
if-system-of-equation-a-x-y-z-a-x-b-y-z-b-and-x-y-c-z-c-is-inconsistent-then-which-of-the-following-is-correct

slug:
if-system-of-equation-a-x-y-z-a-x-b-y-z-b-and-x-y-c-z-c-is-inconsistent-then-which-of-the-following-is-correct-21319

question:
<div class="question">If system of equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mi>a</mi><mo>,</mo><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>c</mi><mi>z</mi><mo>=</mo><mi>c</mi></math> is inconsistent, then which of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>=</mo><mn>1</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>≠</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>≠</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>≠</mo><mn>1</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>≠</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>≠</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>≠</mo><mn>1</mn></math></span> </div>

answer:
<div class="solution"><p>We have, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mi>a</mi></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mi>b</mi></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>ii</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>c</mi><mi>z</mi><mo>=</mo><mi>c</mi></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>iii</mi><mo>)</mo></math></p><p>are inconsistent. </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><mo>∴</mo></mtd><mtd><mfenced close="|" open="|"><mtable columnalign="left"><mtr><mtd><mi>a</mi></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mi>b</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mi>c</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn><mo> </mo><mi>and</mi><mo> </mo><mfenced close="|" open="|"><mtable columnalign="left"><mtr><mtd><mi>a</mi></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi>b</mi></mtd><mtd><mi>b</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mn>1</mn></mtd><mtd><mi>c</mi></mtd></mtr></mtable></mfenced><mo>≠</mo><mn>0</mn></mtd></mtr></mtable></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfenced close="|" open="|"><mtable columnalign="left"><mtr><mtd><mi>a</mi></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mi>b</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mi>c</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>a</mi><mo>(</mo><mi>b</mi><mi>c</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>-</mo><mn>1</mn><mo>(</mo><mi>c</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn><mo>(</mo><mn>1</mn><mo>-</mo><mi>b</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>c</mi><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>-</mo><mi>b</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mtable columnalign="left"><mtr><mtd><mi>a</mi></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi>b</mi></mtd><mtd><mi>b</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mn>1</mn></mtd><mtd><mi>c</mi></mtd></mtr></mtable></mfenced><mo>≠</mo><mn>0</mn></math></p><p>It is possible only when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≠</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>≠</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>≠</mo><mn>1</mn></math></p></div>


title: if-t-0-t-1-t-2-t-n-represent-the-terms-in-the-expansion-of-x-a-n-then-t-0-t-2-t-4-2-t-1-t-3-t-5-2
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
slug:
if-t-0-t-1-t-2-t-n-represent-the-terms-in-the-expansion-of-x-a-n-then-t-0-t-2-t-4-2-t-1-t-3-t-5-2-54924

question:
<div class="question">If $\mathrm{T}_{0}, \mathrm{~T}_{1}, \mathrm{~T}_{2} \ldots \ldots \mathrm{T}_{\mathrm{n}}$ represent the terms in the
<br/> <br/>expansion of $(x+a)^{n}$, then $\left(\mathrm{T}_{0}-\mathrm{T}_{2}+\mathrm{T}_{4}-\ldots \ldots .\right)^{2}+$
<br/> <br/>$\left(\mathrm{T}_{1}-\mathrm{T}_{3}+\mathrm{T}_{5}-\ldots . .\right)^{2}=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(x^{2}+a^{2}\right)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(x^{2}+a^{2}\right)^{n}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(x^{2}+a^{2}\right)^{1 / n}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(x^{2}+a^{2}\right)^{-1 / n}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(x^{2}+a^{2}\right)^{n}$</span> </div>

answer:
<div class="solution">From the given condition, replacing $a$ by a and $-a i$ respectively, we get
<br/> <br/>$(x+a i)^{n}=\left(T_{0}-T_{2}+T_{4}-\ldots \ldots\right)+i\left(T_{1}-T_{3}+T_{5}-\ldots .\right)$ ...(i)
<br/> <br/>and
<br/> <br/>$(x-a i)^{n}=\left(T_{0}-T_{2}+T_{4}-\ldots \ldots\right)-i\left(T_{1}-T_{3}+T_{5}-\ldots\right)$...(ii)
<br/> <br/>Multiplying (ii) and
<br/> <br/>(i) we get required result i.e.,
<br/> <br/>$$
<br/> <br/>\left(x^{2}+a^{2}\right)^{n}=\left(T_{0}-T_{2}+T_{4}-\ldots .\right)^{2}+\left(T_{1}-T_{3}+T_{5}-\ldots\right)^{2}
<br/> <br/>$$</div>


tags: ['Physics', 'Units and Dimensions', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
title:
if-t-2-3-yq-ml-3-then-find-the-dimensions-of-q-where-t-is-the-time-period-of-bar-of-mass-m-length-l-and-young-modulus-y

slug:
if-t-2-3-yq-ml-3-then-find-the-dimensions-of-q-where-t-is-the-time-period-of-bar-of-mass-m-length-l-and-young-modulus-y-47755

question:
<div class="question">If $\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{ML}^{3}}{3 \mathrm{Yq}}}$ then find the dimensions of q. Where $\mathrm{T}$ is the time period of bar of mass $\mathrm{M}$ length $\mathrm{L}$ and Young modulus $\mathrm{Y}$.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$[\mathrm{L}]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{L}^{2}\right]$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{L}^{4}\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{L}^{3}\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{L}^{4}\right]$</span> </div>

answer:
<div class="solution">$\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{ML}^{3}}{3 \mathrm{Yq}}},$ writing dimensions of both<br/>the sides, we get $[\mathrm{T}]=\left[\frac{\mathrm{ML}^{3}}{\mathrm{ML}^{-1} \mathrm{~T}^{-2} \mathrm{q}}\right]^{1 / 2}$<br/>or $\quad q=\left[L^{4}\right]$</div>


title: if-tan-cot-x-cot-tan-x-then-sin-2-x-is-equal-to
slug: if-tan-cot-x-cot-tan-x-then-sin-2-x-is-equal-to-99979
question: <div class="question">If $\tan (\cot x)=\cot (\tan x)$, then $\sin 2 x$ is equal to:</div>
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2}{(2 n+1) \pi}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{4}{(2 n+1) \pi}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{n(n+1) \pi}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{4}{n(n+1) \pi}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4}{(2 n+1) \pi}$</span> </div>

answer:
<div class="solution">Given, $\tan (\cot x)=\cot (\tan x)=\tan \left(\frac{\pi}{2}-\tan x\right)$
<br/> <br/>$\Rightarrow \cot x=n \pi+\frac{\pi}{2}-\tan x$
<br/> <br/>$\Rightarrow \cot x+\tan x=n \pi+\frac{\pi}{2}$
<br/> <br/>$\Rightarrow \frac{1}{\sin x \cos x}=n \pi+\frac{\pi}{2} \Rightarrow \frac{1}{\sin 2 x}=\frac{n \pi}{2}+\frac{\pi}{4}$
<br/> <br/>$\Rightarrow \sin 2 x=\frac{1}{\frac{n \pi}{2}+\frac{\pi}{4}}=\frac{4}{(2 n+1) \pi}$</div>


title: if-tan-mathrmafrac12-and-tan-bfrac13-then-find-the-value-of-mathrmamathrmb
slug: if-tan-mathrmafrac12-and-tan-bfrac13-then-find-the-value-of-mathrmamathrmb-13819
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
question:
<div class="question">If, \(\tan \mathrm{A}=\frac{1}{2}\) and \(\tan B=\frac{1}{3}\), then find the value of \(\mathrm{A}+\mathrm{B}\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\pi\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{\pi}{2}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\frac{\pi}{4}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{-\pi}{4}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{\pi}{4}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2} \cdot \frac{1}{3}}=1 \\<br/>&amp; \therefore A+B=45^{\circ}=\frac{\pi}{4}<br/>\end{aligned}\)</div>


tags: ['Physics', 'Thermal Properties of Matter', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
title:
if-temperature-of-a-black-body-increases-from-300-k-to-900-k-then-the-rate-of-energy-radiation-increases-by

slug:
if-temperature-of-a-black-body-increases-from-300-k-to-900-k-then-the-rate-of-energy-radiation-increases-by-24380

question:
<div class="question">If temperature of a black body increases from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn><mo> </mo><mi mathvariant="normal">K</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>900</mn><mo> </mo><mi mathvariant="normal">K</mi></math>, then the rate of energy radiation increases by</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>81</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>81</mn></math></span> </div>

answer:
<div class="solution"><p>According to Stefan's law <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi><mo>=</mo><mfrac><mi>E</mi><mi>t</mi></mfrac><mo>=</mo><mi>σ</mi><mi>A</mi><mi>e</mi><msup><mi>T</mi><mn>4</mn></msup></math></p><p>So, we can write as</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>E</mi><mn>2</mn></msub><msub><mi>E</mi><mn>1</mn></msub></mfrac><mo>=</mo><msup><mfenced><mfrac><msub><mi>T</mi><mn>2</mn></msub><msub><mi>T</mi><mn>1</mn></msub></mfrac></mfenced><mn>4</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>E</mi><mn>2</mn></msub><msub><mi>E</mi><mn>1</mn></msub></mfrac><mo>=</mo><msup><mfenced><mfrac><mn>900</mn><mn>300</mn></mfrac></mfenced><mn>4</mn></msup><mo>⇒</mo><mfrac><msub><mi>E</mi><mn>2</mn></msub><msub><mi>E</mi><mn>1</mn></msub></mfrac><mo>=</mo><msup><mfenced><mn>3</mn></mfenced><mn>4</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>2</mn></msub><mo>=</mo><mn>81</mn><msub><mi>E</mi><mn>1</mn></msub><mo>  </mo><mo>⇒</mo><mfrac><msub><mi>E</mi><mn>2</mn></msub><msub><mi>E</mi><mn>1</mn></msub></mfrac><mo>=</mo><mn>81</mn></math></p></div>


title: if-the-2-p-th-term-of-a-h-p-is-q-and-the-2-q-th-term-is-p-then-the-2-p-q-th-term-is
slug: if-the-2-p-th-term-of-a-h-p-is-q-and-the-2-q-th-term-is-p-then-the-2-p-q-th-term-is-20935
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the $(2 \mathrm{p})^{\text {th }}$ term of a $\mathrm{H}$. $\mathrm{P}$. is $\mathrm{q}$ and the $(2 \mathrm{q})^{\text {th }}$ term is $\mathrm{p},$ then the $2(\mathrm{p}+\mathrm{q})^{\mathrm{th}}$ term is-</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{p q}{2(p+q)}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2 \mathrm{pq}}{\mathrm{p}+\mathrm{q}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{pq}}{\mathrm{p}+\mathrm{q}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{p}+\mathrm{q}}{\mathrm{pq}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{p}+\mathrm{q}}{\mathrm{pq}}$</span> </div>

answer:
<div class="solution">If a is the first term and d is the common difference of the associated A.P. $\frac{1}{\mathrm{q}}=\frac{1}{\mathrm{a}}+(2 \mathrm{p}-1) \mathrm{d}, \frac{1}{\mathrm{p}}=\frac{1}{\mathrm{a}}+(2 \mathrm{q}-1) \mathrm{d}$
<br/> <br/>$\Rightarrow \mathrm{d}=\frac{1}{2 \mathrm{pq}}$
<br/> <br/>If h is the $2(\mathrm{p}+\mathrm{q})^{\mathrm{th}}$ term $\frac{1}{\mathrm{~h}}=\frac{1}{\mathrm{a}}+(2 \mathrm{p}+2 \mathrm{q}-1) \mathrm{d}$
<br/> <br/>$=\frac{1}{\mathrm{q}}+\frac{1}{\mathrm{p}}=\frac{\mathrm{p}+\mathrm{q}}{\mathrm{pq}}$</div>


tags: ['Physics', 'Magnetic Properties of Matter', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-the-b-h-curves-of-two-samples-of-x-and-y-of-iron-are-as-shown-below-then-which-one-of-the-following-statement-is-correct-1

slug:
if-the-b-h-curves-of-two-samples-of-x-and-y-of-iron-are-as-shown-below-then-which-one-of-the-following-statement-is-correct-1-28170

question:
<div class="question">If the $B-H$ curves of two samples of $X$ and $Y$ of iron are as shown below, then which one of the following statement is correct?
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Xl4sf4ZM6zttiqfH1nkQGUg60x52xUO4PKZCrxH_A-g.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>Both $X$ and $Y$ are suitable for making electromagnets.
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>Both $X$ and $Y$ are suitable for making permanent magnet.
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$X$ is suitable for making permanent magnet and $Y$ for making electromagnet.
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$X$ is suitable for making electromagnet and $Y$ is suitable for permanent magnet.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$X$ is suitable for making permanent magnet and $Y$ for making electromagnet.
<br/></span> </div>

answer:
<div class="solution">For permanent magnet we prefer a material with high retentivity and high coercivity. For electromagnet, we prefer high saturated magnetism, low coercivity and least possible area of hystersis loop. Therefore, $X$ is suitable for making permanent magnet and $Y$ for making electromagnet.</div>


tags: ['Physics', 'Magnetic Properties of Matter', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
if-the-b-h-curves-of-two-samples-of-x-and-y-of-iron-are-as-shown-below-then-which-one-of-the-following-statement-is-correct

slug:
if-the-b-h-curves-of-two-samples-of-x-and-y-of-iron-are-as-shown-below-then-which-one-of-the-following-statement-is-correct-87462

question:
<div class="question">If the $B-H$ curves of two samples of $X$ and $Y$ of iron are as shown below, then which one of the following statement is correct?
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Xl4sf4ZM6zttiqfH1nkQGUg60x52xUO4PKZCrxH_A-g.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>Both $X$ and $Y$ are suitable for making electromagnets.
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>Both $X$ and $Y$ are suitable for making permanent magnet.
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$X$ is suitable for making permanent magnet and $Y$ for making electromagnet.
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$X$ is suitable for making electromagnet and $Y$ is suitable for permanent magnet.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$X$ is suitable for making permanent magnet and $Y$ for making electromagnet.
<br/></span> </div>

answer:
<div class="solution">For permanent magnet we prefer a material with high retentivity and high coercivity. For electromagnet, we prefer high saturated magnetism, low coercivity and least possible area of hystersis loop. Therefore, $X$ is suitable for making permanent magnet and $Y$ for making electromagnet.</div>


tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
if-the-coefficient-of-x-3-and-x-4-in-the-expansion-of-1-a-x-b-x-2-1-2-x-18-in-powers-of-x-are-both-zero-then-a-b-is-equal-to

slug:
if-the-coefficient-of-x-3-and-x-4-in-the-expansion-of-1-a-x-b-x-2-1-2-x-18-in-powers-of-x-are-both-zero-then-a-b-is-equal-to-74693

question:
<div class="question">If the coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> in the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mn>18</mn></msup></math> in powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> are both zero, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></mrow></mfenced></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>16</mn><mo>,</mo><mfrac><mn>251</mn><mn>3</mn></mfrac></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>14</mn><mo>,</mo><mfrac><mn>251</mn><mn>3</mn></mfrac></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>14</mn><mo>,</mo><mfrac><mn>272</mn><mn>3</mn></mfrac></mrow></mfenced></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>16</mn><mo>,</mo><mfrac><mn>272</mn><mn>3</mn></mfrac></mrow></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>16</mn><mo>,</mo><mfrac><mn>272</mn><mn>3</mn></mfrac></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>We have, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mn>18</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>18</mn></mmultiscripts><mn>2</mn><mi>x</mi><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>18</mn></mmultiscripts><mfenced><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mmultiscripts><mi>C</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>18</mn></mmultiscripts><msup><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mn>3</mn></msup><mo>+</mo><mmultiscripts><mi>C</mi><mn>4</mn><none></none><mprescripts></mprescripts><none></none><mn>18</mn></mmultiscripts><msup><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mn>4</mn></msup><mo>…</mo></mrow></mfenced></math></p><p>Coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mmultiscripts><mi>C</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>18</mn></mmultiscripts><msup><mfenced><mn>2</mn></mfenced><mn>3</mn></msup><mo>+</mo><mi>a</mi><mo>·</mo><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>18</mn></mmultiscripts><msup><mfenced><mn>2</mn></mfenced><mn>2</mn></msup><mo>-</mo><msup><mi>b</mi><mn>18</mn></msup><msub><mi>C</mi><mn>1</mn></msub><mfenced><mn>2</mn></mfenced></math></p><p>and coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mmultiscripts><mi>C</mi><mn>4</mn><none></none><mprescripts></mprescripts><none></none><mn>18</mn></mmultiscripts><msup><mfenced><mn>2</mn></mfenced><mn>4</mn></msup><mo>-</mo><mmultiscripts><mi>C</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>18</mn></mmultiscripts><msup><mfenced><mn>2</mn></mfenced><mn>3</mn></msup><mi>a</mi><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>18</mn></mmultiscripts><msup><mfenced><mn>2</mn></mfenced><mn>2</mn></msup><mi>b</mi></math></p><p>Coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> are zero.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo>-</mo><mmultiscripts><mi>C</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>18</mn></mmultiscripts><msup><mfenced><mn>2</mn></mfenced><mn>3</mn></msup><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>18</mn></mmultiscripts><msup><mfenced><mn>2</mn></mfenced><mn>2</mn></msup><mfenced><mi>a</mi></mfenced><mo>-</mo><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>18</mn></mmultiscripts><mfenced><mn>2</mn></mfenced><mi>b</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mrow><mn>4</mn><mo>×</mo><mn>17</mn><mo>×</mo><mn>16</mn></mrow><mrow><mn>3</mn><mo>×</mo><mn>2</mn></mrow></mfrac><mo>-</mo><mn>17</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn><mo>-</mo><mfrac><mn>32</mn><mn>3</mn></mfrac><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced></math></p><p>Solving Eqs. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>ii</mi><mo>)</mo></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>16</mn><mo>,</mo><mi>b</mi><mo>=</mo><mfrac><mn>272</mn><mn>3</mn></mfrac></math></p></div>


tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">(11,2)</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-coordinates-at-one-end-of-a-diameter-of-the-circle-x-2-y-2-8-x-4-y-c-0-are-32-then-the-coordinates-at-the-other-end-are

slug:
if-the-coordinates-at-one-end-of-a-diameter-of-the-circle-x-2-y-2-8-x-4-y-c-0-are-32-then-the-coordinates-at-the-other-end-are-42746

question:
<div class="question">If the coordinates at one end of a diameter of the circle $x^{2}+y^{2}-8 x-4 y+c=0$ are (-3,2) , then the coordinates at the other end are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">(5,3)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">(6,2)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">(1,-8)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">(11,2)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">The centre of the given circle is $\mathrm{C} \equiv(4,2)$
<br/> <br/>Let $A \equiv(-3,2)$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/oPn8SV17oF4gRa73rlmL9sWyWstn65noHV_Lechikcs.original.fullsize.png"/><br/> <br/> <br/>If $(\alpha, \beta)$ are the coordinates of the other end of the diameter, then, as the middle ploint of the diameter is the centre,
<br/> <br/>$\therefore \quad \frac{\alpha-3}{2}=4$ and $\frac{\beta+2}{2}=2 \Rightarrow \alpha=11, \beta=2$
<br/> <br/>Thus, the coordinates of the other end of diameter are (11,2)</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value"><br/>3</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-coordinates-of-the-point-in-which-the-line-joining-the-points-3-5-7-and-2-1-8-is-intersected-by-the-plane-yz-is-a-b-13-c-then-a-b-c

slug:
if-the-coordinates-of-the-point-in-which-the-line-joining-the-points-3-5-7-and-2-1-8-is-intersected-by-the-plane-yz-is-a-b-13-c-then-a-b-c-61847

question:
<div class="question">If the co-ordinates of the point in which the line joining the points $(3,5,-7)$ and $(-2,1,8)$ is intersected by the plane $y z$ is $\left[\mathrm{a}, \frac{13}{\mathrm{~b}}, \mathrm{c}\right]$, then $(a+b-c)=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>2</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>5</span> </li> </ul>

answer:
<div class="solution">Ratio $=-\left(\frac{3}{-2}\right)=\frac{3}{2}$<br/>$\therefore \quad$ Required co-ordinates of the points are<br/>$\begin{aligned}<br/>&amp; {\left[\frac{6-6}{5}, \frac{10+3}{5}, \frac{-14+24}{5}\right]=\left(0, \frac{13}{5}, 2\right) .} \\<br/>&amp; \therefore a+b-c=0+5-2=3<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
title:
if-the-coordinates-of-the-points-a-and-b-be-3-3-and-7-6-then-the-length-of-the-portion-of-the-line-ab-intercepted-between-the-axes-is

slug:
if-the-coordinates-of-the-points-a-and-b-be-3-3-and-7-6-then-the-length-of-the-portion-of-the-line-ab-intercepted-between-the-axes-is-59717

question:
<div class="question">If the coordinates of the points $A$ and $B$ be $(3,3)$ and $(7,6)$, then the length of the portion of the line $\mathrm{AB}$ intercepted between the axes is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{5}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{10}}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{13}}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{4}$</span> </div>

answer:
<div class="solution">Equation of line $A B$ is $y-3=\frac{6-3}{7-3}(x-3)$<br/>$\Rightarrow 3 x-4 y+3=0 \Rightarrow \frac{x}{-1}+\frac{y}{3 / 4}=1$<br/>Hence required length is $\sqrt{(-1)^2+\left(\frac{3}{4}\right)^2}=\frac{5}{4}$.</div>


title: if-the-curve-y-a-x-and-y-b-x-intersect-at-angle-then-tan-is-equal-to
slug: if-the-curve-y-a-x-and-y-b-x-intersect-at-angle-then-tan-is-equal-to-83694
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>b</mi><mi>x</mi></msup></math> intersect at angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mi>α</mi></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>a</mi><mi>b</mi></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>log</mi><mi>a</mi><mo>-</mo><mi>log</mi><mi>b</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>log</mi><mi>a</mi><mi>log</mi><mi>b</mi></mrow></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>a</mi><mi>b</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>log</mi><mi>a</mi><mo>+</mo><mi>log</mi><mi>b</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>log</mi><mi>a</mi><mi>log</mi><mi>b</mi></mrow></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>log</mi><mi>a</mi><mo>-</mo><mi>log</mi><mi>b</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>log</mi><mi>a</mi><mi>log</mi><mi>b</mi></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><p>Clearly, the point of intersection of curves is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math>.</p><p>Now, slope of tangent of first curve</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup><mi>log</mi><mi>a</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mfenced><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mfenced><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>=</mo><msub><mi>m</mi><mn>1</mn></msub><mo>=</mo><mi>log</mi><mi>a</mi></math></p><p>Slope of tangent of second curve,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mn>2</mn></msub><mo>=</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>b</mi><mi>x</mi></msup><mi>log</mi><mi>b</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>  </mo><msub><mi>m</mi><mn>2</mn></msub><mo>=</mo><msub><mfenced><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mfenced><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>=</mo><mi>log</mi><mi>b</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>tan</mi><mi>α</mi><mo>=</mo><mfenced close="|" open="|"><mfrac><mrow><msub><mi>m</mi><mn>1</mn></msub><mo>-</mo><msub><mi>m</mi><mn>2</mn></msub></mrow><mrow><mn>1</mn><mo>+</mo><msub><mi>m</mi><mn>1</mn></msub><msub><mi>m</mi><mn>2</mn></msub></mrow></mfrac></mfenced><mo>=</mo><mfrac><mrow><mi>log</mi><mi>a</mi><mo>-</mo><mi>log</mi><mi>b</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>log</mi><mi>a</mi><mi>log</mi><mi>b</mi></mrow></mfrac></math></p></div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
title:
if-the-distance-between-the-first-maxima-and-fifth-minima-of-a-double-slit-pattern-is-7-mm-and-the-slits-are-separated-by-015-mm-with-the-screen-50-cm

slug:
if-the-distance-between-the-first-maxima-and-fifth-minima-of-a-double-slit-pattern-is-7-mm-and-the-slits-are-separated-by-015-mm-with-the-screen-50-cm-71706

question:
<div class="question">If the distance between the first maxima and fifth minima of a double slit pattern is $7 \mathrm{~mm}$ and the slits are separated by $0.15 \mathrm{~mm}$ with the screen 50 $\mathrm{cm} .$ from the slits, then the wavelength of the light used is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$200 \mathrm{nm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$100 \mathrm{nm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$800 \mathrm{nm}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$600 \mathrm{nm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$600 \mathrm{nm}$</span> </div>

answer:
<div class="solution">There are three and a half fringes from first maxima to fifth minima as shown.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/DHk5IdcNhRtbsrgLZt9dkQBjUw9Nsh1MBaP3byDVApE.original.fullsize.png"/><br/><br/>$\Rightarrow \beta=\frac{7 \mathrm{~mm}}{3.5}=2 \mathrm{~mm} \Rightarrow \lambda=\frac{\beta \mathrm{D}}{\mathrm{d}}=600 \mathrm{nm}$</div>


tags: ['Mathematics', 'Hyperbola', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
title:
if-the-eccentricity-and-length-of-latus-rectum-of-a-hyperbola-are-3-13-and-3-10-units-respectively-then-what-is-the-length-of-the-transverse-axis-1

slug:
if-the-eccentricity-and-length-of-latus-rectum-of-a-hyperbola-are-3-13-and-3-10-units-respectively-then-what-is-the-length-of-the-transverse-axis-1-16315

question:
<div class="question">If the eccentricity and length of latus rectum of a hyperbola are $\frac{\sqrt{13}}{3}$ and $\frac{10}{3}$ units respectively, then what is the length of the transverse axis?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{7}{2}$ unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12 unit</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{15}{2}$ unit</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{15}{4}$ unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{15}{4}$ unit</span> </div>

answer:
<div class="solution">Length of latus rectum of a hyperbola is $\frac{2 b^2}{a}$ where $a$ is the half of the distance between two vertex of the hyperbola.<br/>Latus rectum $=\frac{2 b^2}{a}=\frac{10}{3}$ or, $b^2=\frac{5 a}{3}$ ...(i)<br/>In case of hyperbola, $\mathrm{b}^2=\mathrm{a}^2\left(\mathrm{e}^2-1\right)$ ...(ii)<br/>Putting value of $b^2$ from equation (i) and $e=$<br/>$\frac{\sqrt{13}}{3}$ in equation (ii), $\frac{5 \mathrm{a}}{3}=\mathrm{a}^2\left(\frac{13}{9}-1\right)$ or,<br/>$\frac{5 a}{3}=\frac{4 a^2}{9}$<br/>$\Rightarrow 4 a^2-15 a=0$ or, $a(4-15 a)=0$<br/>$a \neq 0$, hence, $a=\frac{15}{4}$</div>


tags: ['Chemistry', 'Solutions', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
title:
if-the-elevation-in-boiling-point-of-a-solution-of-10-g-of-solute-mol-wt-100-in-100-g-of-water-is-t-b-the-ebullioscopic-constant-of-water-is

slug:
if-the-elevation-in-boiling-point-of-a-solution-of-10-g-of-solute-mol-wt-100-in-100-g-of-water-is-t-b-the-ebullioscopic-constant-of-water-is-36114

question:
<div class="question">If the elevation in boiling point of a solution of $10 \mathrm{~g}$ of solute $(\mathrm{mol} . \mathrm{wt}=100)$ in $100 \mathrm{~g}$ of water is $\Delta T b$, the ebullioscopic constant of water is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\Delta T_{b}}{10}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\Delta T_{b}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10 \Delta T_{b}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$100 \Delta T_{b}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\Delta T_{b}$</span> </div>

answer:
<div class="solution">$\mathrm{m}=\frac{1000 \times \mathrm{k}_{b} \times \mathrm{w}}{W \times \Delta T_{b}}$
<br/>or $\quad \begin{aligned} k_{\mathrm{b}} &amp;=\frac{\mathrm{m} \times W \times \Delta T_{b}}{1000 \times \mathrm{w}}=\frac{100 \times 100 \times \Delta T_{b}}{1000 \times 10} \\ &amp;=\Delta T_{b} \end{aligned}$</div>


title: if-the-energy-of-a-photon-is-10-ev-then-its-momentum-is
slug: if-the-energy-of-a-photon-is-10-ev-then-its-momentum-is-81173
question: <div class="question">If the energy of a photon is $10 \mathrm{eV}$, then its momentum is</div>
tags: ['Physics', 'Dual Nature of Matter', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5.33 \times 10^{-23} \mathrm{~kg} \mathrm{~m} / \mathrm{s}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5.33 \times 10^{-25} \mathrm{~kg} \mathrm{~m} / \mathrm{s}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5.33 \times 10^{-29} \mathrm{~kg} \mathrm{~m} / \mathrm{s}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$5.33 \times 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5.33 \times 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}$</span> </div>

answer:
<div class="solution">Momentum of a photon $\propto \frac{\mathrm{E}}{\mathrm{c}}$<br/>$$<br/>=5.33 \times 10^{-27} \mathrm{~kg} \mathrm{~ms}^{-1}<br/>$$</div>


title: if-the-enthalpy-of-vaporization-of-water-is-1865-kjmol-1-the-entropy-if-its-vaporization-will-be
tags: ['Chemistry', 'Thermodynamics (C)', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
slug:
if-the-enthalpy-of-vaporization-of-water-is-1865-kjmol-1-the-entropy-if-its-vaporization-will-be-84734

question:
<div class="question">If the enthalpy of vaporization of water is 186.5 $\mathrm{kJmol}^{-1}$, the entropy if its vaporization will be:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0.5 \mathrm{kJK}^{-1} \mathrm{~mol}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1.0 \mathrm{kJK}^{-1} \mathrm{~mol}^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1.5 \mathrm{kJK}^{-1} \mathrm{~mol}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.0 \mathrm{kJK}^{-1} \mathrm{~mol}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.5 \mathrm{kJK}^{-1} \mathrm{~mol}^{-1}$</span> </div>

answer:
<div class="solution">Given enthalpy of vaporization,
<br/> <br/>$$
<br/> <br/>\Delta \mathrm{H}=186.5 \mathrm{~kJ} \mathrm{~mol}^{-1}
<br/> <br/>$$
<br/> <br/>Boiling point of water
<br/> <br/>$$
<br/> <br/>=100^{\circ} \mathrm{C}=100+273=373 \mathrm{~K}
<br/> <br/>$$
<br/> <br/>Entropy change,
<br/> <br/>$$
<br/> <br/>\begin{array}{r}
<br/> <br/>\Delta \mathrm{S}=\frac{\Delta \mathrm{H}}{\mathrm{T}}=\frac{186.5 \mathrm{~kJ} \mathrm{~mol}^{-1}}{373 \mathrm{~K}} \\
<br/> <br/>=0.5 \mathrm{~kJ} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Mathematics', 'Parabola', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
title:
if-the-equations-of-the-pairs-of-opposite-sides-of-a-parallelogram-are-x25-x60-and-y26-y50-then-equations-of-its-diagonals-are

slug:
if-the-equations-of-the-pairs-of-opposite-sides-of-a-parallelogram-are-x25-x60-and-y26-y50-then-equations-of-its-diagonals-are-60437

question:
<div class="question">If the equations of the pairs of opposite sides of a parallelogram are \(x^2-5 x+6=0\) and \(y^2-6 y+5=0\), then equations of its diagonals are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(x+4 y=13, y=4 x-7\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(4 x+y=13,4 y=x-7\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(4 x+y=13, y=4 x-7\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(y-4 x=13, y+4 x=7\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(4 x+y=13, y=4 x-7\)</span> </div>

answer:
<div class="solution">Equations of the sides of the parallelogram are<br/>\((x-3)(x-2)=0\) and \((y-5)(y-1)=0\)<br/>i.e. \(x=3, x=2 ; y=5, y=1\)<br/>Hence its vertices are : \(\mathrm{A}(2,1) ; \mathrm{B}(3,1)\); \(\mathrm{C}(3,5) ; \mathrm{D}(2,5)\)<br/>Equation of the diagonal \(\mathrm{AC}\) is<br/>\(y-1=\frac{4}{1}(x-2) \Rightarrow y=4 x-7\)<br/>Equation of the diagonal \(\mathrm{BD}\) is<br/>\(y-1=\frac{4}{-1}(x-3) \Rightarrow 4 x+y=13\)</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-equilibrium-constant-of-the-reaction-2-hi-h-2-i-2-is-025-then-the-equilibrium-constant-for-the-reaction-h-2-i-2-2-hi-would-be

slug:
if-the-equilibrium-constant-of-the-reaction-2-hi-h-2-i-2-is-025-then-the-equilibrium-constant-for-the-reaction-h-2-i-2-2-hi-would-be-97861

question:
<div class="question">If the equilibrium constant of the reaction $2 \mathrm{HI} \rightleftharpoons \mathrm{H}_{2}+\mathrm{I}_{2}$ is $0.25,$ then the equilibrium constant for the reaction $\mathrm{H}_{2}+\mathrm{I}_{2} \rightleftharpoons 2 \mathrm{HI}$ would be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">When the reaction is reversed,<br/>$$<br/>\mathrm{K}^{\prime}=\frac{1}{\mathrm{~K}}=\frac{1}{0.25}=4<br/>$$</div>


tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
title:
if-the-expression-x211-xa-and-x214-x2-a-must-have-a-common-factor-and-mathrma-neq-0-then-the-common-factor-is

slug:
if-the-expression-x211-xa-and-x214-x2-a-must-have-a-common-factor-and-mathrma-neq-0-then-the-common-factor-is-30913

question:
<div class="question">If the expression \(x^2-11 x+a\) and \(x^2-14 x+2 a\) must have a common factor and \(\mathrm{a} \neq 0\), then, the common factor is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\((x-3)\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\((x-6)\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\((x-8)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((x-8)\)</span> </div>

answer:
<div class="solution">Here Let \(x-\alpha\) is the common factor then \(\mathrm{x}=\alpha\) is root of the corresponding equation \(\therefore \alpha^2-11 \alpha+a=0\)<br/>\(\alpha^2-14 \alpha+2 \mathrm{a}=0\)<br/>Subtracting \(3 \alpha-a=0 \Rightarrow \alpha=a / 3\)<br/>Hence \(\frac{a^2}{9}-11 \frac{a}{3}+a=0, a=0\) or \(a=24\) since \(\mathrm{a} \neq 0, \mathrm{a}=24\)<br/>\(\therefore\) the common factor of \(\left\{\begin{array}{l}x^2-11 x+24 \\ x^2-14 x+48\end{array}\right.\) is clearly \(x -8\)</div>


title: if-the-force-is-given-by-f-at-bt-2-with-t-as-time-the-dimensions-of-a-and-b-are
slug: if-the-force-is-given-by-f-at-bt-2-with-t-as-time-the-dimensions-of-a-and-b-are-61582
tags: ['Physics', 'Units and Dimensions', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the force is given by $\mathrm{F}=\mathrm{at}+\mathrm{bt}^{2}$ with $\mathrm{t}$ as time. The dimensions of a and b are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{MLT}^{-4}\right]$ and $\left[\mathrm{MLT}^{-2}\right]$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{MLT}^{-3}\right]$ and $\left[\mathrm{MLT}^{-4}\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{ML}^{2} \mathrm{~T}^{-3}\right]$ and $\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{ML}^{2} \mathrm{~T}^{-3}\right]$ and $\left[\mathrm{ML}^{3} \mathrm{~T}^{-4}\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{MLT}^{-3}\right]$ and $\left[\mathrm{MLT}^{-4}\right]$</span> </div>

answer:
<div class="solution">Dimension of at = Dimension of F 
<br/>$[a t]=[F] \Rightarrow[a]=\left[\frac{F}{t}\right]$
<br/>$[b]=\left[\frac{M L T^{-2}}{T}\right] \Rightarrow[a]=\left[M L T^{-3}\right]$
<br/>Dimension of $\mathrm{bt}^{2}=$ Dimension of $\mathrm{F}$
<br/>$\left[\mathrm{bt}^{2}\right]=[\mathrm{F}] \Rightarrow[\mathrm{b}]=\left[\frac{\mathrm{F}}{\mathrm{t}^{2}}\right]$
<br/>$[\mathrm{b}]=\left[\frac{\mathrm{ML} \mathrm{T}^{-4}}{\mathrm{~T}^{2}}\right] \Rightarrow[\mathrm{b}]=\left[\mathrm{MLT}^{-4}\right]$</div>


tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
title:
if-the-forward-voltage-in-a-semiconductor-diode-is-changed-from-05-v-to-07-v-then-the-forward-current-changes-by-10-ma-the-forward-resistance-of-diode

slug:
if-the-forward-voltage-in-a-semiconductor-diode-is-changed-from-05-v-to-07-v-then-the-forward-current-changes-by-10-ma-the-forward-resistance-of-diode-76765

question:
<div class="question">If the forward voltage in a semiconductor diode is changed from $0.5 \mathrm{~V}$ to $0.7 \mathrm{~V}$, then the forward current changes by $1.0 \mathrm{~mA}$. The forward resistance of diode junction will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$100 \Omega$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$120 \Omega$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$200 \Omega$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$240 \Omega$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$200 \Omega$</span> </div>

answer:
<div class="solution">Forward resistance
<br/> <br/>$=\frac{\Delta \mathrm{V}}{\Delta \mathrm{I}}=\frac{0.7-0.5}{1.0 \times 10^{-3}}=200 \Omega$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3,-5</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-function-f-x-a-x-b-3-x-b-x-2-a-x-1-1-x-2-x-2-is-continuous-at-x-2-and-4-then-the-values-of-a-and-b-are

slug:
if-the-function-f-x-a-x-b-3-x-b-x-2-a-x-1-1-x-2-x-2-is-continuous-at-x-2-and-4-then-the-values-of-a-and-b-are-99645

question:
<div class="question">If the function $f(x)=\begin{cases}A x-B, &amp; x \leq 1 \\ 3 x &amp; , 1<x b="" x=""></x>is continuous at $x=2$ and $4,$ then the values of a and $b$ are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3,5</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">3,-5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0,3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0,5</span> </li> </ul>

answer:
<div class="solution">Since $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=2$
<br/>$\therefore \mathrm{f}(2)=\lim _{\mathrm{x} \rightarrow 2^{+}} \mathrm{f}(\mathrm{x}) \Rightarrow 1=\lim _{\mathrm{x} \rightarrow 2^{+}}(\mathrm{ax}+\mathrm{b})$
<br/>$\therefore 1=2 \mathrm{a}+\mathrm{b}$
<br/>Again $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=4$
<br/>$\therefore \mathrm{f}(4)=\lim _{x \rightarrow 4^{-}} \mathrm{f}(\mathrm{x}) \Rightarrow 7=\lim _{\mathrm{x} \rightarrow 4^{-}}(\mathrm{ax}+\mathrm{b})$
<br/>$\therefore 7=4 \mathrm{a}+\mathrm{b}$
<br/>Solving (1) and $(2),$ we get $a=3, b=-5$</div>


title: if-the-function-finfty-infty-rightarrow-b-defined-by-fxx26-x8-is-bijective-then-b
slug: if-the-function-finfty-infty-rightarrow-b-defined-by-fxx26-x8-is-bijective-then-b-70040
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the function \(f:(-\infty, \infty) \rightarrow B\) defined by \(f(x)=-x^2+6 x-8\) is bijective, then \(B=\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\([1, \infty)\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\((-\infty, 1]\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\((-\infty, \infty)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((-\infty, 1]\)</span> </div>

answer:
<div class="solution">Since the function \(f\) is bijective, therefore \(f\) is onto. Therefore range of \(f=B\).<br/>Let \(y=-x^2+6 x-8\)<br/>\(\Rightarrow \mathrm{x}^2-6 \mathrm{x}+(8+\mathrm{y})=0\)<br/>\(\Rightarrow x=\frac{6 \pm \sqrt{36-4(8+y)}}{2}=\frac{6 \pm \sqrt{4-4 y}}{2}\)<br/>For \(x\) to be real, \(4-4 y \geq 0 \Rightarrow y \leq 1\)<br/>\(\therefore \mathrm{B}=\) range of \(\mathrm{F}=(-\infty, 1]\)</div>


title: if-the-harmonic-mean-between-a-and-b-be-h-then-the-value-of-h-a-1-h-b-1-is
slug: if-the-harmonic-mean-between-a-and-b-be-h-then-the-value-of-h-a-1-h-b-1-is-90395
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the harmonic mean between $a$ and $b$ be $H$, then the value of $\frac{1}{\mathrm{H}-a}+\frac{1}{\mathrm{H}-b}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a+b$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a b$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{a}+\frac{1}{b}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{a}-\frac{1}{b}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{a}+\frac{1}{b}$</span> </div>

answer:
<div class="solution">Putting $\mathrm{H}=\frac{2 a b}{a+b}$, we have
<br/> <br/>$\begin{array}{l}
<br/> <br/>\frac{1}{\mathrm{H}-a}+\frac{1}{\mathrm{H}-b} \\
<br/> <br/>=\frac{1}{\left(\frac{2 a b}{a+b}-a\right)}+\frac{1}{\left(\frac{2 a b}{a+b}-b\right)}=\frac{a+b}{a b-a^{2}}+\frac{a+b}{a b-b^{2}}
<br/> <br/>\end{array}$
<br/> <br/>$=\left(\frac{a+b}{b-a}\right)\left(\frac{1}{a}-\frac{1}{b}\right)=\left(\frac{a+b}{b-a}\right)\left(\frac{b-a}{a b}\right)=\frac{a+b}{a b}=\frac{1}{a}+\frac{1}{b}$</div>


tags: ['Mathematics', 'Ellipse', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
if-the-length-of-the-major-axis-of-the-ellipse-x-2-a-2-y-2-b-2-1-is-three-times-the-length-of-minor-axis-its-eccentricity-is

slug:
if-the-length-of-the-major-axis-of-the-ellipse-x-2-a-2-y-2-b-2-1-is-three-times-the-length-of-minor-axis-its-eccentricity-is-44683

question:
<div class="question">If the length of the major axis of the ellipse <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>x</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn></math> is three times the length of minor axis, its eccentricity is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>We have</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>x</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn></math></p><p>Given,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Length of the major axis</mtext><mo>=</mo><mn>3</mn><mfenced><mtext>Length of the minor axis</mtext></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced><mrow><mn>2</mn><mi>a</mi></mrow></mfenced><mo>=</mo><mn>3</mn><mfenced><mrow><mn>2</mn><mi>b</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><mn>3</mn><mi>b</mi></math></p><p>Now,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><msqrt><mn>1</mn><mo>-</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>e</mi><mo>=</mo><msqrt><mn>1</mn><mo>-</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mn>9</mn><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>e</mi><mo>=</mo><msqrt><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>9</mn></mfrac></msqrt><mo>=</mo><mfrac><msqrt><mn>8</mn></msqrt><mn>3</mn></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac></math></p></div>


title: if-the-line-2-x-3-y-k-touches-the-parabola-y-2-6-x-then-find-the-value-of-k
slug: if-the-line-2-x-3-y-k-touches-the-parabola-y-2-6-x-then-find-the-value-of-k-45222
tags: ['Mathematics', 'Parabola', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the line $2 \mathrm{x}-3 \mathrm{y}=\mathrm{k}$ touches the parabola $\mathrm{y}^{2}=6 \mathrm{x},$ then find the value of $\mathrm{k}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-15 / 4$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-27 / 4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-1 / 4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-3 / 4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-27 / 4$</span> </div>

answer:
<div class="solution">$$
<br/> <br/>\text {  } \begin{aligned}
<br/> <br/>\text { Given } x=\frac{3 y+k}{2} &amp; \ldots \ldots . . \\
<br/> <br/>\text { and } y^{2}=6 x &amp; \\
<br/> <br/>\Rightarrow y^{2}=6\left(\frac{3 y+k}{2}\right) \\
<br/> <br/>\Rightarrow y^{2}=3(3 y+k) \Rightarrow y^{2}-9 y-3 k=0 &amp; \ldots \ldots . .
<br/> <br/>\end{aligned}
<br/> <br/>$$
<br/> <br/>If line (1) touches parabola (2) then roots of quadratic equation ( 3 ) is equal $\therefore(-9)^{2}=4 \times 1 \times(-3 k) \Rightarrow k=-27 / 4$</div>


title: if-the-line-2-x10-is-the-directrix-of-the-parabola-y2k-x60-then-one-of-the-values-of-k-is
slug: if-the-line-2-x10-is-the-directrix-of-the-parabola-y2k-x60-then-one-of-the-values-of-k-is-39748
tags: ['Mathematics', 'Ellipse', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-6</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the line \(2 x-1=0\) is the directrix of the parabola \(y^2-k x+6=0\), then one of the values of \(k\) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">-6</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(1 / 4\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(-1 / 4\)</span> </li> </ul>

answer:
<div class="solution">Given equation of parabola is \(y^2-k x+6=0\)<br/>\(\Rightarrow y^2=k x-6 \Rightarrow y^2=k\left(x-\frac{6}{k}\right)\)<br/>Now, directrix, \(x-\frac{6}{k}=-\frac{k}{4}\)<br/>\(\Rightarrow x=\frac{6}{k}-\frac{k}{4}\) ...(i)<br/>But directrix is given \(\Rightarrow x=\frac{1}{2}\) ...(ii)<br/>\(\begin{aligned}<br/>&amp; \therefore \frac{6}{k}-\frac{k}{4}=\frac{1}{2} \Rightarrow 24-k^2=2 k \\<br/>&amp; \Rightarrow k^2+2 k-24=0 \\<br/>&amp; \Rightarrow(k+6)(k-4)=0 \quad \Rightarrow k=-6, k=4<br/>\end{aligned}\)</div>


title: if-the-line-fracx41fracy21fraczk2-lies-in-the-plane-2-x4-yz7-then-the-value-of-k-is
slug: if-the-line-fracx41fracy21fraczk2-lies-in-the-plane-2-x4-yz7-then-the-value-of-k-is-49595
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the line \(\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-k}{2}\) lies in the plane \(2 x-4 y+z=7\), then the value of \(k\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-7</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">No real value</span> </li> </ul>

answer:
<div class="solution">The point \((4,2, k)\) on the line also lies on the plane \(2 x-4 y+z=7\)<br/>So, \(8-8+\mathrm{k}=7 \Rightarrow \mathrm{k}=7\)</div>


tags: ['Mathematics', 'Circle', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-the-line-x-cos-y-sin-p-represents-the-common-chord-of-the-circles-x-2-y-2-a-2-and-x-2-y-2-b-2-a-b-where-a-and-b-lie-ont-he-first-circle-and-p-and-q-1

slug:
if-the-line-x-cos-y-sin-p-represents-the-common-chord-of-the-circles-x-2-y-2-a-2-and-x-2-y-2-b-2-a-b-where-a-and-b-lie-ont-he-first-circle-and-p-and-q-1-37214

question:
<div class="question">If the line $x \cos \alpha+y \sin \alpha=p$ represents the common chord of the circles $x^{2}+y^{2}=a^{2}$ and $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{b}^{2}(\mathrm{a}&gt;\mathrm{b}),$ where $\mathrm{A}$ and $\mathrm{B}$ lie ont he first circle and $\mathrm{P}$ and $\mathrm{Q}$ lie on the second circle, then AP is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{a^{2}+p^{2}}+\sqrt{b^{2}+p^{2}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{a^{2}-p^{2}}+\sqrt{b^{2}-p^{2}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{a^{2}-p^{2}}-\sqrt{b^{2}-p^{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{a^{2}+p^{2}}-\sqrt{b^{2}+p^{2}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{a^{2}-p^{2}}-\sqrt{b^{2}-p^{2}}$</span> </div>

answer:
<div class="solution">The given circles are concentric with centre at (0,0) and the length of the perpendicular from (0,0) on the given line is $\mathrm{p}$. Let $\mathrm{OL}=\mathrm{p}$
<br/>then, $\mathrm{AL}=\sqrt{\mathrm{OA}^{2}-\mathrm{OL}^{2}}=\sqrt{\mathrm{a}^{2}-\mathrm{p}^{2}}$
<br/>and $\mathrm{PL}=\sqrt{\mathrm{OP}^{2}-\mathrm{OL}^{2}}=\sqrt{\mathrm{b}^{2}-\mathrm{p}^{2}}$
<br/>$\Rightarrow \mathrm{AP}=\sqrt{\mathrm{a}^{2}-\mathrm{p}^{2}}-\sqrt{\mathrm{b}^{2}-\mathrm{p}^{2}}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/I7c8MVgqVnzEIH0N6EFDLp_CwP9Wef5DEKrZcfzvT_Y.original.fullsize.png"/></div>


tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
title:
if-the-line-x-cos-y-sin-p-represents-the-common-chord-of-the-circles-x-2-y-2-a-2-and-x-2-y-2-b-2-a-b-where-a-and-b-lie-ont-he-first-circle-and-p-and-q

slug:
if-the-line-x-cos-y-sin-p-represents-the-common-chord-of-the-circles-x-2-y-2-a-2-and-x-2-y-2-b-2-a-b-where-a-and-b-lie-ont-he-first-circle-and-p-and-q-58865

question:
<div class="question">If the line $x \cos \alpha+y \sin \alpha=p$ represents the common chord of the circles $x^{2}+y^{2}=a^{2}$ and $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{b}^{2}(\mathrm{a}&gt;\mathrm{b}),$ where $\mathrm{A}$ and $\mathrm{B}$ lie ont he first circle and $\mathrm{P}$ and $\mathrm{Q}$ lie on the second circle, then AP is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{a^{2}+p^{2}}+\sqrt{b^{2}+p^{2}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{a^{2}-p^{2}}+\sqrt{b^{2}-p^{2}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{a^{2}-p^{2}}-\sqrt{b^{2}-p^{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{a^{2}+p^{2}}-\sqrt{b^{2}+p^{2}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{a^{2}-p^{2}}-\sqrt{b^{2}-p^{2}}$</span> </div>

answer:
<div class="solution">The given circles are concentric with centre at (0,0) and the length of the perpendicular from (0,0) on the given line is $\mathrm{p}$. Let $\mathrm{OL}=\mathrm{p}$
<br/>then, $\mathrm{AL}=\sqrt{\mathrm{OA}^{2}-\mathrm{OL}^{2}}=\sqrt{\mathrm{a}^{2}-\mathrm{p}^{2}}$
<br/>and $\mathrm{PL}=\sqrt{\mathrm{OP}^{2}-\mathrm{OL}^{2}}=\sqrt{\mathrm{b}^{2}-\mathrm{p}^{2}}$
<br/>$\Rightarrow \mathrm{AP}=\sqrt{\mathrm{a}^{2}-\mathrm{p}^{2}}-\sqrt{\mathrm{b}^{2}-\mathrm{p}^{2}}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/I7c8MVgqVnzEIH0N6EFDLp_CwP9Wef5DEKrZcfzvT_Y.original.fullsize.png"/></div>


title: if-the-lines-3-x4-y40-and-6-x8-y70-are-tangents-to-a-circle-then-the-radius-of-the-circle-is
slug: if-the-lines-3-x4-y40-and-6-x8-y70-are-tangents-to-a-circle-then-the-radius-of-the-circle-is-90129
tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the lines \(3 x-4 y+4=0\) and \(6 x-8 y-7=0\) are tangents to a circle, then the radius of the circle is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(3 / 2\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(3 / 4\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(1 / 10\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(1 / 20\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(3 / 4\)</span> </div>

answer:
<div class="solution">The diameter of the circle is perpendicular distance between the parallel lines (tangents) \(3 x-4 y+4=0\) and \(3 x-4 y-\frac{7}{2}=0\) and so it is equal to<br/>\(\frac{4}{\sqrt{9+16}}+\frac{7 / 2}{\sqrt{9+16}}=\frac{3}{2}\). Hence radius is \(\frac{3}{4}\)</div>


tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
title:
if-the-lines-p-1-x-q-1-y-1-p-2-x-q-2-y-1-and-p-3-x-q-3-y-1-be-concurrent-then-the-points-p-1-q-1-p-2-q-2-and-p-3-q-3

slug:
if-the-lines-p-1-x-q-1-y-1-p-2-x-q-2-y-1-and-p-3-x-q-3-y-1-be-concurrent-then-the-points-p-1-q-1-p-2-q-2-and-p-3-q-3-81424

question:
<div class="question">If the lines $\mathrm{p}_{1} \mathrm{x}+\mathrm{q}_{1} \mathrm{y}=1, \mathrm{p}_{2} \mathrm{x}+\mathrm{q}_{2} \mathrm{y}=1$ and $\mathrm{p}_{3} \mathrm{x}+ \mathrm{q}_{3} \mathrm{y}=1$ be concurrent, then the points $\left(\mathrm{p}_{1}, \mathrm{q}_{1}\right),$
<br/> <br/>$\left(p_{2}, q_{2}\right)$ and $\left(p_{3}, q_{3}\right)$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">are collinear</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">form an equilateral triangle</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">form a sealene triangle</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">form a right angled triangle</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">are collinear</span> </div>

answer:
<div class="solution">The equations of the lines are
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\mathrm{p}_{1} \mathrm{x}+\mathrm{q}_{1} \mathrm{y}-1=0 \\
<br/> <br/>\mathrm{p}_{2} \mathrm{x}+\mathrm{q}_{2} \mathrm{y}-1=0
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>and $\mathrm{p}_{3} \mathrm{x}+\mathrm{q}_{3} \mathrm{y}-1=0$
<br/> <br/>As they are concurrent,
<br/> <br/>$$
<br/> <br/>\left|\begin{array}{lll}
<br/> <br/>\mathrm{p}_{1} &amp; \mathrm{q}_{1} &amp; -1 \\
<br/> <br/>\mathrm{p}_{2} &amp; \mathrm{q}_{2} &amp; -1 \\
<br/> <br/>\mathrm{p}_{3} &amp; \mathrm{q}_{3} &amp; -1
<br/> <br/>\end{array}\right|=0 \Rightarrow\left|\begin{array}{lll}
<br/> <br/>\mathrm{p}_{1} &amp; \mathrm{q}_{1} &amp; 1 \\
<br/> <br/>\mathrm{p}_{2} &amp; \mathrm{q}_{2} &amp; 1 \\
<br/> <br/>\mathrm{p}_{3} &amp; \mathrm{q}_{3} &amp; 1
<br/> <br/>\end{array}\right|=0
<br/> <br/>$$
<br/> <br/>This is also the condition for the points $\left(\mathrm{p}_{1}, \mathrm{q}_{1}\right),\left(\mathrm{p}_{2}, \mathrm{q}_{2}\right)$ and $\left(\mathrm{p}_{3}, \mathrm{q}_{3}\right)$ to be collinear.</div>


title: if-the-lines-x-my-n-0-mx-ny-0-and-nx-y-m-0-are-concurrent-then
slug: if-the-lines-x-my-n-0-mx-ny-0-and-nx-y-m-0-are-concurrent-then-74554
tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the lines $\ell \mathrm{x}+\mathrm{my}+\mathrm{n}=0, \mathrm{mx}+\mathrm{ny}+\ell=0$ and
<br/> <br/>$\mathrm{nx}+\ell \mathrm{y}+\mathrm{m}=0$ are concurrent then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\ell+m+n=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\ell-m-n=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\ell+m-n=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$m+n-\ell=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\ell+m+n=0$</span> </div>

answer:
<div class="solution">Since the lines are concurrent, so
<br/> <br/>$$
<br/> <br/>\Rightarrow \begin{array}{l}
<br/> <br/>\left|\begin{array}{lll}
<br/> <br/>\ell &amp; \mathrm{m} &amp; \mathrm{n} \\
<br/> <br/>\mathrm{m} &amp; \mathrm{n} &amp; \ell \\
<br/> <br/>\mathrm{n} &amp; \ell &amp; \mathrm{m}
<br/> <br/>\end{array}\right|=0 \Rightarrow 3 \ell \mathrm{mn}-\ell^{3}-\mathrm{m}^{3}-\mathrm{n}^{3}=0 \\
<br/> <br/>(\ell+\mathrm{m}+\mathrm{n})\left(\ell^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}-(\mathrm{m}-\mathrm{mn}-\mathrm{n} \ell)=0\right.
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>$\Rightarrow \quad \ell+\mathrm{m}+\mathrm{n}=0\left[\because \ell^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}&gt;6 \mathrm{~m}+\mathrm{mn}+\mathrm{n}\right]$</div>


title: if-the-matrix-1-2-3-3-4-5-2-8-10-is-singular-then
slug: if-the-matrix-1-2-3-3-4-5-2-8-10-is-singular-then-48630
tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the matrix $\left[\begin{array}{ccc}1 &amp; 3 &amp; \lambda+2 \\ 2 &amp; 4 &amp; 8 \\ 3 &amp; 5 &amp; 10\end{array}\right]$ is singular, then $\lambda=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-2</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-4</span> </li> </ul>

answer:
<div class="solution">$|\mathrm{A}|=0$ as the matrix $\mathrm{A}$ is singular
<br/> <br/>$$
<br/> <br/>\therefore|\mathrm{A}|=\left|\begin{array}{ccc}
<br/> <br/>1 &amp; 3 &amp; \lambda+2 \\
<br/> <br/>2 &amp; 4 &amp; 8 \\
<br/> <br/>3 &amp; 5 &amp; 10
<br/> <br/>\end{array}\right|=0
<br/> <br/>$$
<br/> <br/>Apply $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-2 \mathrm{R}_{1}$ and $\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-3 \mathrm{R}_{1}$
<br/> <br/>and expand. $-2(4-3 \lambda)+4(4-2 \lambda)=0$
<br/> <br/>$$
<br/> <br/>\Rightarrow 8-2 \lambda=0 \Rightarrow \lambda=4
<br/> <br/>$$
<br/> <br/>For $\lambda=4,$ the second and the third column are proportional.</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
title:
if-the-mean-mode-and-sd-of-a-frequency-distribution-are-41-45-and-8-respectively-then-its-pearsons-coefficient-of-skewness-is

slug:
if-the-mean-mode-and-sd-of-a-frequency-distribution-are-41-45-and-8-respectively-then-its-pearsons-coefficient-of-skewness-is-84341

question:
<div class="question">If the mean, mode and S.D. of a frequency distribution are 41, 45 and 8 respectively, then its Pearson's coefficient of skewness is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{1}{3}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{-1}{2}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{2}{3}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{-1}{2}\)</span> </div>

answer:
<div class="solution">\(-\frac{1}{2}\), We have \(\mathrm{S}_{\mathrm{K}}=\frac{\text { Mean }- \text { Mode }}{\mathrm{S} . \mathrm{D} \text {. }}\)<br/>\(\frac{41-45}{8}=-\frac{1}{2}\)</div>


tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
title:
if-the-middle-points-of-sides-bc-ca-ab-of-triangle-abc-are-respectively-d-e-f-then-position-vector-of-centre-of-triangle-def-when-position-vector-of-a

slug:
if-the-middle-points-of-sides-bc-ca-ab-of-triangle-abc-are-respectively-d-e-f-then-position-vector-of-centre-of-triangle-def-when-position-vector-of-a-49721

question:
<div class="question">If the middle points of sides $\mathrm{BC}, \mathrm{CA} \&amp; \mathrm{AB}$ of triangle $\mathrm{ABC}$ are respectively $\mathrm{D}, \mathrm{E}, \mathrm{F}$ then position vector of centre of triangle $\mathrm{DEF}$, when position vector of $A, \quad B, \quad C$ are respectively $\hat{i}+\hat{j}, \hat{j}+\hat{k}, \hat{k}+\hat{i}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3}(\hat{i}+\hat{j}+\hat{k})$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(\hat{i}+\hat{j}+\hat{k})$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2(\hat{i}+\hat{j}+\hat{k})$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{2}{3}(\hat{i}+\hat{j}+\hat{k})$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2}{3}(\hat{i}+\hat{j}+\hat{k})$</span> </div>

answer:
<div class="solution">The position vector of points $D, E, F$ are respectively $\hat{i}+\hat{j}+\hat{k}, \hat{i}+\frac{\hat{k}+\hat{j}}{2}$ and $\frac{\hat{i}+\hat{k}}{2}+\hat{j}$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/0a-Uy1XmUWvPVWLHzrK4nTHUIIX4kCwelvlDzp_CzjA.original.fullsize.png"/><br/> <br/> <br/>So, position vector of centre of $\Delta \mathrm{DEF}$ $=\frac{1}{3}\left[\frac{\hat{i}+\hat{j}}{2}+\hat{k}+\hat{i} \frac{\hat{k}+\hat{j}}{2}+\frac{\hat{i}+\hat{k}}{2}+\hat{j}\right]$
<br/> <br/>$=\frac{2}{3}[\hat{i}+\hat{j}+\hat{k}]$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
title:
if-the-middle-points-of-sides-mathrmbc-mathrmca-mathrmab-of-triangle-a-b-c-are-respectively-d-e-f-then-position-vector-of-centre-of-triangle-mathrmdef

slug:
if-the-middle-points-of-sides-mathrmbc-mathrmca-mathrmab-of-triangle-a-b-c-are-respectively-d-e-f-then-position-vector-of-centre-of-triangle-mathrmdef-67427

question:
<div class="question">If the middle points of sides \(\mathrm{BC}, \mathrm{CA} ~\&amp;~ \mathrm{AB}\) of triangle \(A B C\) are respectively \(D, E, F\) then position vector of centre of triangle \(\mathrm{DEF}\), when position vector of \(A, B, C\) are respectively \(\hat{i}+\hat{j}\), \(\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{k}}+\hat{\mathrm{i}}\) is -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\((1 / 3)(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\((\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(2(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\((2 / 3)(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((2 / 3)(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})\)</span> </div>

answer:
<div class="solution">The position vector of points \(D, E, F\) are respectively<br/>\(\frac{\mathrm{i}+\mathrm{j}}{2}+\mathrm{k}, \mathrm{i}+\frac{\mathrm{k}+\mathrm{j}}{2} \text { and } \frac{\mathrm{i}+\mathrm{k}}{2}+\mathrm{j}\)<br/>So, position vector of centre of \(\triangle \mathrm{DEF}\)<br/>\(=\frac{1}{3}\left[\frac{\mathrm{i}+\mathrm{j}}{2}+\mathrm{k}+\mathrm{i} \frac{\mathrm{k}+\mathrm{j}}{2}+\frac{\mathrm{i}+\mathrm{k}}{2}+\mathrm{j}\right]=\frac{2}{3}[\mathrm{i}+\mathrm{j}+\mathrm{k}]\)</div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-normal-to-the-curve-y-f-x-at-the-point-34-makes-an-angle-3-4-with-the-positive-x-axis-then-f-3

slug:
if-the-normal-to-the-curve-y-f-x-at-the-point-34-makes-an-angle-3-4-with-the-positive-x-axis-then-f-3-93789

question:
<div class="question">If the normal to the curve $y=f(x)$ at the point (3,4) makes an angle $3 \pi / 4$ with the positive $\mathrm{x}$ -axis, then $\mathrm{f}^{\prime}(3)=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-3 / 4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 / 3$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Slope of normal to $y=f(x)$ at (3,4) is $\frac{-1}{f^{\prime}(3)}$.
<br/>$\text { Thus, } \begin{array}{c}
<br/>\frac{-1}{\mathrm{f}^{\prime}(3)}=\tan \left(\frac{3 \pi}{4}\right)=\tan \left(\frac{\pi}{2}+\frac{\pi}{4}\right) \\
<br/>=-\cot \frac{\pi}{4}=-1 \Rightarrow \mathrm{f}^{\prime}(3)=1
<br/>\end{array}$</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
title:
if-the-papers-of-4-students-can-be-checked-by-any-one-of-the-7-teachers-then-the-probability-that-all-the-4-papers-are-checked-by-exactly-2-teachers-i

slug:
if-the-papers-of-4-students-can-be-checked-by-any-one-of-the-7-teachers-then-the-probability-that-all-the-4-papers-are-checked-by-exactly-2-teachers-i-96037

question:
<div class="question">If the papers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> students can be checked by any one of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> teachers, then the probability that all the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> papers are checked by exactly <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> teachers, is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>12</mn><mn>49</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>6</mn><mn>49</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>9</mn><mn>49</mn></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>15</mn><mn>49</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>6</mn><mn>49</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>Total ways in which papers can be checked is equal <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>7</mn><mn>4</mn></msup></math>. Now, two teachers who have to check all the papers can be selected in <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts></math> ways and papers can be checked by them is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mn>2</mn><mn>4</mn></msup><mo>-</mo><mn>2</mn></mrow></mfenced></math> favourable ways.</p><p>Thus, required probability</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><mfenced><mrow><msup><mn>2</mn><mn>4</mn></msup><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><msup><mn>7</mn><mn>4</mn></msup></mfrac><mo>=</mo><mfrac><mn>6</mn><mn>49</mn></mfrac></math></p></div>


tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
if-the-photon-of-wavelength-150-pm-strikes-an-atom-and-one-of-its-inner-bound-electron-is-ejected-out-with-a-velocity-of-1-5-10-7-m-s-what-is-the-ener

slug:
if-the-photon-of-wavelength-150-pm-strikes-an-atom-and-one-of-its-inner-bound-electron-is-ejected-out-with-a-velocity-of-1-5-10-7-m-s-what-is-the-ener-33777

question:
<div class="question">If the photon of wavelength <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>150</mn><mo> </mo><mi>pm</mi></math> strikes an atom and one of its inner bound electron is ejected out with a velocity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>7</mn></msup><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math>, what is the energy with which it is bound to the nucleus?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>2</mn></msup><mo> </mo><mi>eV</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>15</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo> </mo><mi>eV</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo> </mo><mi>eV</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>12</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo> </mo><mi>eV</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo> </mo><mi>eV</mi></math></span> </div>

answer:
<div class="solution"><blockquote><p>Total energy <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">E</mi></mfenced></math> of any photon is given by the relation : <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">E</mi><mo>=</mo><mfrac><mi>hc</mi><mi mathvariant="normal">λ</mi></mfrac></math></p><p>Where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">h</mi><mo>=</mo></math> Planck's constant <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>6</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>34</mn></mrow></msup><mo> </mo><mi>Js</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">c</mi><mo>=</mo></math> Velocity of light <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">λ</mi><mo>=</mo></math> wavelength <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>10</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi><mo>,</mo></math> i.e, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>150</mn><mo> </mo><mi>pm</mi></mrow></mfenced></math></p><p>Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">E</mi><mo>=</mo><mfrac><mi>hc</mi><mi mathvariant="normal">λ</mi></mfrac><mo>=</mo><mfrac><mrow><mn>6</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>34</mn></mrow></msup><mo>×</mo><mn>3</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup></mrow><mrow><mn>1</mn><mo>.</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>10</mn></mrow></msup></mrow></mfrac><mo>;</mo><mo> </mo><mi mathvariant="normal">E</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>32</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>15</mn></mrow></msup><mo> </mo><mi mathvariant="normal">J</mi></math> and, energy of ejected electron <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msup><mi mathvariant="normal">E</mi><mo>'</mo></msup></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">E</mi><mo>'</mo></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>mv</mi><mn>2</mn></msup></math>,</p><p>where,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">m</mi><mo>=</mo></math> mass of electron <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>9</mn><mo>.</mo><mn>1</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>31</mn></mrow></msup><mo> </mo><mi>kg</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">v</mi><mo>=</mo></math> velocity of electron</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>7</mn></msup><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">E</mi><mo>'</mo></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>mv</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>9</mn><mo>.</mo><mn>1</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>31</mn></mrow></msup><mo>×</mo><msup><mfenced><mrow><mn>1</mn><mo>.</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>7</mn></msup></mrow></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">E</mi><mo>'</mo></msup><mo>=</mo><mn>1</mn><mo>.</mo><mn>024</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>16</mn></mrow></msup></math></p><p>Thus, total energy of photon <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math> Binding energy of electron</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">B</mi></mfenced></math> + energy of ejected electron <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msup><mi mathvariant="normal">E</mi><mo>'</mo></msup></mfenced></math></p><p>Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>32</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>15</mn></mrow></msup><mo>=</mo><mfenced><mi mathvariant="normal">B</mi></mfenced><mo>+</mo><mfenced><mi mathvariant="normal">E</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfenced><mi mathvariant="normal">B</mi></mfenced><mo>=</mo><mfenced><mi mathvariant="normal">E</mi></mfenced><mo>-</mo><mfenced><msup><mi mathvariant="normal">E</mi><mo>'</mo></msup></mfenced><mo>=</mo><mfenced close="]" open="["><mrow><mn>1</mn><mo>.</mo><mn>32</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>15</mn></mrow></msup></mrow></mfenced><mo>-</mo><mfenced close="]" open="["><mrow><mn>1</mn><mo>.</mo><mn>024</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>16</mn></mrow></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>2176</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>15</mn></mrow></msup><mo> </mo><mi mathvariant="normal">J</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>2176</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>15</mn></mrow></msup></mrow><mrow><mn>1</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>19</mn></mrow></msup></mrow></mfrac><mo> </mo><mi>eV</mi><mo>=</mo><mn>7</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo> </mo><mi>eV</mi></math></p></blockquote></div>


title: if-the-plane-3-x-y-2-z-6-0-is-parallel-to-the-line-2-b-3-x-1-3-y-a-z-1-then-the-value-of-3-a-3-b-is
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
slug:
if-the-plane-3-x-y-2-z-6-0-is-parallel-to-the-line-2-b-3-x-1-3-y-a-z-1-then-the-value-of-3-a-3-b-is-50909

question:
<div class="question">If the plane $3 x+y+2 z+6=0$ is parallel to the line $\frac{3 x-1}{2 b}=3-y=\frac{z-1}{a}$, then the value of $3 a+3 b$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$\frac{1}{2}$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$\frac{3}{2}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>3
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>4</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$\frac{3}{2}$
<br/></span> </div>

answer:
<div class="solution">Given plane $3 x+y+2 z+6=0$
<br/>and line $\frac{x-\frac{1}{3}}{\frac{2 b}{3}}=\frac{y-3}{-1}=\frac{z-1}{a}$
<br/>Since, plane is parallel to line, then
<br/>$$
<br/>\begin{aligned}
<br/>&amp; 3\left(\frac{2 b}{3}\right)+(1)(-1)+2(a)=0 \\
<br/>&amp; 2 b-1+2 a=0 \\
<br/>&amp; \Rightarrow \quad a+b=\frac{1}{2} \\
<br/>&amp;
<br/>\end{aligned}
<br/>$$
<br/>Now, $3 a+3 b=\frac{3}{2}$</div>


title: if-the-planes-r-2-i-j-3-k-0-and-r-i-5-j-k-5-are-perpendicular-to-each-other-then-the-value-of-2-is
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
slug:
if-the-planes-r-2-i-j-3-k-0-and-r-i-5-j-k-5-are-perpendicular-to-each-other-then-the-value-of-2-is-44644

question:
<div class="question">If the planes <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>r</mi><mo>→</mo></mover><mo>·</mo><mfenced><mrow><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mi mathvariant="normal">λ</mi><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></mrow></mfenced><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>r</mi><mo>→</mo></mover><mo>·</mo><mfenced><mrow><mi mathvariant="normal">λ</mi><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>5</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>-</mo><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></mrow></mfenced><mo>=</mo><mn>5</mn></math> are perpendicular to each other, then the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>2</mn></msup><mo>+</mo><mi>λ</mi></math> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>r</mi><mo>→</mo></mover><mo>·</mo><mfenced><mrow><mi mathvariant="normal">λ</mi><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>5</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>-</mo><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></mrow></mfenced><mo>=</mo><mn>5</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>r</mi><mo>→</mo></mover><mo>·</mo><mfenced><mrow><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mi mathvariant="normal">λ</mi><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></mrow></mfenced><mo>=</mo><mn>0</mn></math></p><p>We know that two planes <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>r</mi><mo>→</mo></mover><mo>·</mo><msub><mover><mi>n</mi><mo>→</mo></mover><mn>1</mn></msub><mo>=</mo><msub><mi>d</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>r</mi><mo>→</mo></mover><mo>·</mo><msub><mover><mi>n</mi><mo>→</mo></mover><mn>2</mn></msub><mo>=</mo><msub><mi>d</mi><mn>2</mn></msub></math> are perpendicular if</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>n</mi><mo>→</mo></mover><mn>1</mn></msub><mo>·</mo><msub><mover><mi>n</mi><mo>→</mo></mover><mn>2</mn></msub><mo>=</mo><mn>0</mn></math></p><p>Therefore,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi mathvariant="normal">λ</mi><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>5</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>-</mo><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></mrow></mfenced><mo>·</mo><mfenced><mrow><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mi mathvariant="normal">λ</mi><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></mrow></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>5</mn><mi>λ</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>-</mo><mn>3</mn><mi>λ</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>λ</mi><mo>=</mo><mo>-</mo><mn>1</mn></math></p><p>Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>2</mn></msup><mo>+</mo><mi>λ</mi><mo>=</mo><mn>0</mn></math></p></div>


tags: ['Physics', 'Capacitance', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
title:
if-the-potential-of-a-capacitor-having-capacity-6-f-is-increased-from-10-v-to-20-v-then-increase-in-its-energywill-be

slug:
if-the-potential-of-a-capacitor-having-capacity-6-f-is-increased-from-10-v-to-20-v-then-increase-in-its-energywill-be-84819

question:
<div class="question">If the potential of a capacitor having capacity $6 \mu \mathrm{F}$ is increased from $10 \mathrm{~V}$ to $20 \mathrm{~V}$, then increase in its energywill be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4 \times 10^{-4} \mathrm{~J}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \times 10^{-6} \mathrm{~J}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$9 \times 10^{-4} \mathrm{~J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$12 \times 10^{-6} \mathrm{~J}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$9 \times 10^{-4} \mathrm{~J}$</span> </div>

answer:
<div class="solution">Capacitance of capacitor (C) $=6 \mu \mathrm{F}$ $=6 \times 10^{-6} \mathrm{~F}$; Initial potential $\left(\mathrm{V}_{1}\right)=10 \mathrm{~V}$ and<br/>final potential $\left(\mathrm{V}_{2}\right)=20 \mathrm{~V}$.<br/>The increase in energy $(\Delta \mathrm{U})$<br/>$$<br/>\begin{array}{l}<br/>=\frac{1}{2} \mathrm{C}\left(\mathrm{V}_{2}^{2}-\mathrm{V}_{1}^{2}\right) \\<br/>=\frac{1}{2} \times\left(6 \times 10^{-6}\right) \times\left[(20)^{2}-(10)^{2}\right] \\<br/>=\left(3 \times 10^{-6}\right) \times 300=9 \times 10^{-4} \mathrm{~J} .<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
title:
if-the-radius-of-a-sphere-is-measured-as-9-cm-with-an-error-of-003-cm-then-find-the-approximating-error-in-calculating-its-volume

slug:
if-the-radius-of-a-sphere-is-measured-as-9-cm-with-an-error-of-003-cm-then-find-the-approximating-error-in-calculating-its-volume-11863

question:
<div class="question">If the radius of a sphere is measured as $9 \mathrm{~cm}$ with an error of $0.03 \mathrm{~cm}$, then find the approximating error in calculating its volume.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.46 \pi \mathrm{cm}^{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$8.62 \pi \mathrm{cm}^{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$9.72 \pi \mathrm{cm}^{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$7.6 \pi \mathrm{cm}^{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$9.72 \pi \mathrm{cm}^{3}$</span> </div>

answer:
<div class="solution">Let $r$ be the radius of the sphere and $\Delta r$ be the error in measuring the radius. Then, $r=9 \mathrm{~cm}$ and $\Delta r=0.03 \mathrm{~cm}$
<br/> <br/>Let $V$ be the volume of the sphere. Then,
<br/> <br/>$\begin{array}{l}
<br/> <br/>\mathrm{V}=\frac{4}{3} \pi r^{3} \\
<br/> <br/>\Rightarrow \frac{\mathrm{dV}}{\mathrm{dr}}=4 \pi r^{2} \\
<br/> <br/>\Rightarrow\left(\frac{\mathrm{dV}}{\mathrm{dr}}\right)_{r=9}=4 \pi \times 9^{2}=324 \pi
<br/> <br/>\end{array}$
<br/> <br/>Let $\Delta \mathrm{V}$ be the error in $\mathrm{V}$ due to error $\Delta \mathrm{r}$ in $r$.
<br/> <br/>Then,
<br/> <br/>$\begin{array}{l}
<br/> <br/>\Delta \mathrm{V}=\frac{\mathrm{dV}}{\mathrm{dr}} \Delta r \\
<br/> <br/>\Rightarrow \Delta \mathrm{V}=324 \pi \times 0.03=9.72 \pi \mathrm{cm}^{3}
<br/> <br/>\end{array}$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
title:
if-the-radius-of-a-star-is-r-and-it-acts-as-a-black-body-what-would-be-the-temperature-of-the-star-in-which-the-rate-of-energy-production-is-q

slug:
if-the-radius-of-a-star-is-r-and-it-acts-as-a-black-body-what-would-be-the-temperature-of-the-star-in-which-the-rate-of-energy-production-is-q-61777

question:
<div class="question">If the radius of a star is $R$ and it acts as a black body, what would be the temperature of the star, in which the rate of energy production is $Q$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$Q / 4 \pi R^{2} \sigma$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\mathrm{Q} / 4 \pi R^{2} \sigma\right)^{-1 / 2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(4 \pi R^{2} Q / \sigma\right)^{1 / 4}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\left(\mathrm{Q} / 4 \pi R^{2} \sigma\right)^{1 / 4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\mathrm{Q} / 4 \pi R^{2} \sigma\right)^{1 / 4}$</span> </div>

answer:
<div class="solution">Stefan's law for black bodyradiation $Q=\sigma e A T^{4}$
<br/> <br/>$$
<br/> <br/>T=\left[\frac{Q}{\sigma\left(4 \pi R^{2}\right)}\right]^{1 / 4}
<br/> <br/>$$
<br/> <br/>Here $e=1$
<br/> <br/>$$
<br/> <br/>A=4 \pi R^{2}
<br/> <br/>$$</div>


title: if-the-radius-of-h-is-053-a-then-what-will-be-the-radius-of-3-li-2
slug: if-the-radius-of-h-is-053-a-then-what-will-be-the-radius-of-3-li-2-10718
tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the radius of $\mathrm{H}$ is $0.53 Ãâ¦$, then what will be the radius of ${ }_{3} \mathrm{Li}^{2+}$?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0.17 Ãâ¦$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.36 Ãâ¦$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.53 Ãâ¦$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.59 Ãâ¦$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.17 Ãâ¦$</span> </div>

answer:
<div class="solution">Radius of orbit $=\frac{\mathrm{n}^{2} \mathrm{a}_{\mathrm{o}}}{\mathrm{z}}\left(\mathrm{a}_{\mathrm{o}}=0.529 \mathrm{~A}\right)$
<br/>Radius of $\mathrm{H}=\frac{(1)^{2} \times 0.529 \mathrm{~A}}{1}=0.53 \mathrm{~A}$
<br/>Thus, the radius of ${ }_{3} \mathrm{Li}^{2+}$ will be $=\frac{(1)^{2} \times 0.529}{3}=0.17 Ãâ¦$</div>


tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
title:
if-the-radius-of-the-3-rd-bohrs-orbit-of-hydrogen-atom-is-r-3-and-the-radius-of-4-th-bohrs-orbit-is-r-4-then

slug:
if-the-radius-of-the-3-rd-bohrs-orbit-of-hydrogen-atom-is-r-3-and-the-radius-of-4-th-bohrs-orbit-is-r-4-then-11738

question:
<div class="question">If the radius of the $3 \mathrm{rd}$ Bohr's orbit of hydrogen atom is $r_3$ and the radius of $4^{\text {th }}$ Bohr's orbit is $r_4$. Then:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$r_4=\frac{9}{16} r_3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$r_4=\frac{16}{9} r_3$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$r_4=\frac{3}{4} r_3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$r_4=\frac{4}{3} r_3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$r_4=\frac{16}{9} r_3$</span> </div>

answer:
<div class="solution"><br/>$\begin{aligned} &amp; r=0.529 \times \frac{n^2}{Z} Ã \\ &amp; r_3=0.529 \times \frac{3^2}{1} \Rightarrow r_4=0.529 \times \frac{4^2}{1} \\ &amp; \frac{r_4}{r_3}=\frac{4^2}{3^2}=\frac{16}{9} \Rightarrow r_4=\frac{16 r_3}{9}\end{aligned}$</div>


tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
title:
if-the-radius-of-the-earth-were-to-shrink-by-one-per-cent-its-mass-remaining-the-same-the-value-of-g-on-the-earths-surface-would

slug:
if-the-radius-of-the-earth-were-to-shrink-by-one-per-cent-its-mass-remaining-the-same-the-value-of-g-on-the-earths-surface-would-74655

question:
<div class="question">If the radius of the earth were to shrink by one per cent, its mass remaining the same, the value of g on the earth's surface would</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">increase by $0.5 \%$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">increase by $2 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">decrease by $0.5 \%$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">decrease by $2 \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">increase by $2 \%$</span> </div>

answer:
<div class="solution">$\quad \mathrm{g}=\frac{\mathrm{GM}}{\mathrm{R}^{2}} \Rightarrow \frac{\mathrm{dg}}{\mathrm{g}}=-2 \frac{\mathrm{dR}}{\mathrm{R}}$
<br/>$\frac{\mathrm{dR}}{\mathrm{R}}=-1 \% \Rightarrow \frac{\mathrm{dg}}{\mathrm{g}}=2 \%$</div>


tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
title:
if-the-real-part-of-z-1-z-2-is-4-z-1-then-the-locus-of-the-point-representing-z-in-the-complex-plane-is

slug:
if-the-real-part-of-z-1-z-2-is-4-z-1-then-the-locus-of-the-point-representing-z-in-the-complex-plane-is-72189

question:
<div class="question">If the real part of $\frac{\bar{z}+2}{\bar{z}-1}$ is $4, z \neq 1,$ then the locus of the point representing $z$ in the complex plane is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a straight line parallel to $x$ -axis</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a straight line equally inclined to axes</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a circle with radius 2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">a circle with radius $\frac{1}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a circle with radius $\frac{1}{2}$</span> </div>

answer:
<div class="solution">Real part of $\frac{\bar{z}+2}{\bar{z}-1}$ is given by
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\frac{1}{2}\left[\frac{\bar{z}+2}{\bar{z}-1}+\left(\frac{\bar{z}+2}{\bar{z}-1}\right)\right]=4 \\
<br/> <br/>\Rightarrow \frac{\bar{z}+2}{\bar{z}-1}+\frac{z+2}{z-1}=8 \\
<br/> <br/>\Rightarrow z \bar{z}-\bar{z}+2 z-2+z \bar{z}+2 \bar{z}-z-2 \\
<br/> <br/>=8(z \bar{z}-\bar{z}-z+1) \\
<br/> <br/>\Rightarrow z \bar{z}-\frac{3}{2} z-\frac{3}{2} \bar{z}+2=0
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Comparing with the equation $z \bar{z}+\bar{a} z+a \bar{z}+b=0,$ we get $a=-\frac{3}{2}$ and $b$
<br/> <br/>$=2$. Thus, the locus of $z$ given by the equation
<br/> <br/>(i) is a circle with centre $=\frac{1}{2}$</div>


tags: ['Physics', 'Atomic Physics', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
title:
if-the-series-limit-wavelength-of-lyman-series-for-the-hydrogen-atom-is-912-a-then-the-series-limit-wavelength-for-balmer-series-of-hydrogen-atoms-is

slug:
if-the-series-limit-wavelength-of-lyman-series-for-the-hydrogen-atom-is-912-a-then-the-series-limit-wavelength-for-balmer-series-of-hydrogen-atoms-is-34195

question:
<div class="question">If the series limit wavelength of Lyman series for the hydrogen atom is $912 Ã,$ then the series limit wavelength for Balmer series of hydrogen atoms is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$912 Ã$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$912 \times 2 Ã$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$912 \times 4 Ã$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{912}{2} Ã$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$912 \times 4 Ã$</span> </div>

answer:
<div class="solution">$\frac{1}{\lambda}=R\left[\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right]$
<br/> <br/>For limiting wavelength of Lyman series
<br/> <br/>$$
<br/> <br/>n_{1}=1, n_{2}=\infty \quad \frac{1}{\lambda_{L}}=R
<br/> <br/>$$
<br/> <br/>For limiting wavelength of Balmer series $n_{1}=2, n_{2}=\infty$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\frac{1}{\lambda_{B}}=R\left(\frac{1}{4}\right) \Rightarrow \lambda_{B}=\frac{4}{R} \\
<br/> <br/>\therefore \lambda_{B}=4 \lambda_{L}=4 \times 912 \AA
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
title:
if-the-slope-of-the-tangent-to-the-curve-at-any-point-p-x-y-is-x-y-cos-2-x-y-then-the-equation-of-a-curve-passing-through-1-4-is

slug:
if-the-slope-of-the-tangent-to-the-curve-at-any-point-p-x-y-is-x-y-cos-2-x-y-then-the-equation-of-a-curve-passing-through-1-4-is-85983

question:
<div class="question">If the slope of the tangent to the curve at any point $\mathrm{P}(x, y)$ is $\frac{y}{x}-\cos ^{2} \frac{y}{x}$, then the equation of a curve passing through $\left(1, \frac{\pi}{4}\right)$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\tan \left(\frac{y}{x}\right)+\log x=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\tan \left(\frac{y}{x}\right)+\log y=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\tan \left(\frac{x}{y}\right)+\log x=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\tan \left(\frac{x}{y}\right)+\log y=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan \left(\frac{y}{x}\right)+\log x=1$</span> </div>

answer:
<div class="solution">According to the condition,
<br/> <br/>$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{y}{x}-\cos ^{2} \frac{y}{x} \ldots \text { (i) }$
<br/> <br/>This is a homogeneous differential equation Substituting $y=v x$, we get
<br/> <br/>$\begin{array}{l}
<br/> <br/>v+x \frac{\mathrm{dv}}{\mathrm{dx}}=v-\cos ^{2} v \\
<br/> <br/>\Rightarrow x \frac{\mathrm{dv}}{\mathrm{dx}}=-\cos ^{2} v \\
<br/> <br/>\Rightarrow \int \sec ^{2} \mathrm{vdv}=-\int \frac{\mathrm{dx}}{\mathrm{x}} \\
<br/> <br/>\Rightarrow \tan v=-\log x+\mathrm{C} \\
<br/> <br/>\Rightarrow \tan \frac{y}{x}+\log x=\mathrm{C}
<br/> <br/>\end{array}$
<br/> <br/>Substituting $\mathrm{x}=1, y=\frac{\pi}{4}$, we get $\mathrm{C}=1$
<br/> <br/>Thus, we get
<br/> <br/>$\tan \left(\frac{y}{x}\right)+\log x=1$
<br/> <br/>which is the required solution.</div>


title: if-the-straight-line-y-m-x-c-touches-the-parabola-y-2-4-a-x-4-a-3-0-then-c-is
slug: if-the-straight-line-y-m-x-c-touches-the-parabola-y-2-4-a-x-4-a-3-0-then-c-is-11108
tags: ['Mathematics', 'Parabola', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the straight line $y=m x+c$ touches the parabola $y^2-4 a x+4 a^3=0$, then $c$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$a m+\frac{a}{m}$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$a m-\frac{a}{m}$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$\frac{a}{m}+a^2 m$
<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"> <br/>$\frac{a}{m}-a^2 m$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$\frac{a}{m}-a^2 m$</span> </div>

answer:
<div class="solution">Solving the given equations,
<br/>$$
<br/>\begin{aligned}
<br/>&amp;(m x+c)^2=4 a x-4 a^3 \\
<br/>&amp; \Rightarrow m^2 x^2+2 m c \cdot x+c^2=4 a x-4 a^3 \\
<br/>&amp; \Rightarrow m^2 x^2+(2 m c-4 a) x+c^2+4 a^3=0
<br/>\end{aligned}
<br/>$$
<br/>Since the straight line touches the parabola at a point, so, the discriminant $=0$
<br/>$$
<br/>\begin{aligned}
<br/>&amp; \Rightarrow \quad(2 m c-4 a)^2-4 m^2\left(c^2+4 a^3\right)=0 \\
<br/>&amp; \Rightarrow 4 m^2 c^2-16 a m c+16 a^2-4 m^2 c^2-16 a^3 m^2=0 \\
<br/>&amp; \Rightarrow-m c+a-a^2 m^2=0 \\
<br/>&amp; \Rightarrow m c=a-a^2 m^2 \Rightarrow c=\frac{a}{m}-a^2 m
<br/>\end{aligned}
<br/>$$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
title:
if-the-straight-lines-k-x-1-2-y-2-3-z-3-and-3-x-2-k-y-3-2-z-1-intersect-at-a-point-then-the-integer-k-is-equal-to-1

slug:
if-the-straight-lines-k-x-1-2-y-2-3-z-3-and-3-x-2-k-y-3-2-z-1-intersect-at-a-point-then-the-integer-k-is-equal-to-1-86213

question:
<div class="question">If the straight lines $\frac{x-1}{k}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x-2}{3}=\frac{y-3}{k}=\frac{z-1}{2}$ intersect at a point, then the integer $\mathrm{k}$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>-5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>-2</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>-5</span> </div>

answer:
<div class="solution">When the two lines intersect then shortes distance between them is zero i.e.<br/>$\begin{aligned}<br/>&amp; \frac{\left(\vec{a}_2-\vec{a}_1\right) \cdot \vec{b}_1 \times \vec{b}_2}{\left|\vec{b}_1 \times \vec{b}_2\right|}=0 \\<br/>&amp; \Rightarrow\left(\vec{a}_2-\vec{a}_1\right) \cdot \vec{b}_1 \times \vec{b}_2=0<br/>\end{aligned}$<br/>where $\vec{a}_1=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}_1=k \hat{i}+2 \hat{j}+3 \hat{k}$<br/>$\begin{aligned}<br/>&amp; \vec{a}_2=2 \hat{i}+3 \hat{j}+\hat{k}, \hat{b}_2=3 \hat{i}+k \hat{j}+2 \hat{k} \\<br/>&amp; \Rightarrow\left|\begin{array}{ccc}<br/>1 &amp; 1 &amp; -2 \\<br/>k &amp; 2 &amp; 3 \\<br/>3 &amp; k &amp; 2<br/>\end{array}\right|=0 \\<br/>&amp; \Rightarrow \quad 1(4-3 k)-1(2 k-9)-2\left(k^2-6\right)=0 \\<br/>&amp; \Rightarrow \quad-2 k^2-5 k+25=0 \Rightarrow k=-5 \text { or } \frac{5}{2}<br/>\end{aligned}$</div>


title: if-the-sum-of-a-certain-number-of-terms-of-the-ap-25-22-19-is-116-then-the-last-term-is
slug: if-the-sum-of-a-certain-number-of-terms-of-the-ap-25-22-19-is-116-then-the-last-term-is-35581
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the sum of a certain number of terms of the A.P. $25,22,19, \ldots$ is 116 then the last term is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution">$a=25, d=22-25=-3 \text {. }$
<br/> <br/>Let $n$ be the no. of terms
<br/> <br/>$\begin{array}{l}
<br/> <br/>\text { Sum }=116 ; \\
<br/> <br/>\text { Sum }=\frac{n}{2}[2 a+(n-1) d] \\
<br/> <br/>116=\frac{n}{2}[50+(n-1)(-3)] \\
<br/> <br/>\text { or } 232=n[50-3 n+3]=n[53-3 n]=-3 n^{2}+53 n \\
<br/> <br/>\Rightarrow 3 n^{2}-53+232=0 \\
<br/> <br/>\Rightarrow(n-8)(3 n-29)=0 \\
<br/> <br/>\Rightarrow n=8 \text { or } n=\frac{29}{3}, \\
<br/> <br/>n \neq \frac{29}{3} \\
<br/> <br/>\therefore n=8 \\
<br/> <br/>\therefore \text { Now, } \mathrm{T}_{8}=a+(8-1) d=25+7 \times(-3)=25-21=4 \\
<br/> <br/>\therefore \text { Last term }=4
<br/> <br/>\end{array}$</div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
title:
if-the-sum-of-an-infinite-gp-a-a-r-a-r-2-a-r-3-is-15-and-the-sum-of-the-squares-of-its-each-term-is-150-then-the-sum-of-a-r-2-a-r-4-a-r-6-is-1

slug:
if-the-sum-of-an-infinite-gp-a-a-r-a-r-2-a-r-3-is-15-and-the-sum-of-the-squares-of-its-each-term-is-150-then-the-sum-of-a-r-2-a-r-4-a-r-6-is-1-80689

question:
<div class="question">If the sum of an infinite GP $a, a r, a r^2, a r^3, \ldots$ is 15 and the sum of the squares of its each term is 150 , then the sum of $a r^2, a r^4, a r^6, \ldots$ is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{5}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{25}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{9}{2}$</span> </li> </ul>

answer:
<div class="solution">Sum of infinite terms of series a $+a r+a r^2+$ $=15$<br/>$\therefore \frac{a}{1-r}=15$ ...(i)<br/>Sequence formed by square of terms: $a^2, a^2 r^2, a^2 r^4, a^2 r^6 \ldots$<br/>$a^2+a^2 r^2+\cdots=150$<br/>Sum $=\frac{a^2}{1-r^2}=150$<br/>$\Rightarrow \frac{a}{1-r} \cdot \frac{a}{1+r}=15 \Rightarrow 15 \cdot \frac{a}{1+r}=150$ $\left(\because \frac{\mathrm{a}}{1-\mathrm{r}}=15\right)$<br/>$\Rightarrow \frac{a}{1+r}=10$ ...(ii)<br/>On dividing equation (i) by (ii)<br/>$\frac{1+\mathrm{r}}{1-\mathrm{r}}=\frac{15}{10}$<br/>or $r=\frac{1}{5} \Rightarrow a=12$<br/>Now series : $a r^2+a r^4+a r^6+\ldots$<br/>Sum $=\frac{a^2}{1-r^2}=\frac{12 \cdot\left(\frac{1}{25}\right)}{1-\frac{1}{25}}=\frac{1}{2}$</div>


tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
title:
if-the-sum-of-odd-numbered-terms-and-the-sum-of-even-numbered-terms-in-the-expansion-of-x-a-n-are-a-and-b-respectively-then-the-value-of-x-2-a-2-n-is

slug:
if-the-sum-of-odd-numbered-terms-and-the-sum-of-even-numbered-terms-in-the-expansion-of-x-a-n-are-a-and-b-respectively-then-the-value-of-x-2-a-2-n-is-41894

question:
<div class="question">If the sum of odd numbered terms and the sum of even numbered terms in the expansion of $(x+a)^{n}$ are $A$ and $B$ respectively, then the value of $\left(x^{2}-\right.$ $\left.a^{2}\right)^{n}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{A}^{2}-\mathrm{B}^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{A}^{2}+\mathrm{B}^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 \mathrm{AB}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{A}^{2}-\mathrm{B}^{2}$</span> </div>

answer:
<div class="solution">$$
<br/> <br/>\begin{array}{r}
<br/> <br/>(x+a)^{n}={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1} a+{ }^{n} C_{2} x^{n-2} a^{2} \\
<br/> <br/>+{ }^{n} C_{3} x^{n-3} a^{3}+{ }^{n} C_{4} x^{n-4} a^{4}+\ldots . \\
<br/> <br/>=\left({ }^{n} C_{0} x^{n}+{ }^{n} C_{2} x^{n-2} a^{2}+{ }^{n} C_{4} x^{n-4} a^{4}+\ldots .\right)+ \\
<br/> <br/>+\left({ }^{n} C_{1} x^{n-1} a+{ }^{n} C_{3} x^{n-3} a^{3}+{ }^{n} C_{5} x^{n-5} a^{5}\right)+\ldots .
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>$=A+B$ ....(i)
<br/> <br/>Similarly, $(x-a)^{n}=A-B$ ....(ii)
<br/> <br/>Multiplying eqns. (1) and (2), we get
<br/> <br/>$$
<br/> <br/>\left(\mathrm{x}^{2}-\mathrm{a}^{2}\right)^{\mathrm{n}}=\mathrm{A}^{2}-\mathrm{B}^{2}
<br/> <br/>$$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
if-the-sum-of-squares-of-distances-of-a-point-from-the-planes-x-y-z-0-x-z-0-and-x-2-y-z-0-is-p-2-then-locus-of-the-point-is

slug:
if-the-sum-of-squares-of-distances-of-a-point-from-the-planes-x-y-z-0-x-z-0-and-x-2-y-z-0-is-p-2-then-locus-of-the-point-is-25307

question:
<div class="question">If the sum of squares of distances of a point from the planes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>0</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup></math> then locus of the point is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><msup><mi>p</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><msup><mi>p</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><msup><mi>p</mi><mn>2</mn></msup></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><msup><mi>p</mi><mn>2</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><msup><mi>p</mi><mn>2</mn></msup></math></span> </div>

answer:
<div class="solution"><p>Let distances of a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></mfenced></math> from the planes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>0</mn></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi></mrow><msqrt><mn>3</mn></msqrt></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mi>x</mi><mo>-</mo><mi>z</mi></mrow><msqrt><mn>2</mn></msqrt></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi></mrow><msqrt><mn>6</mn></msqrt></mfrac></math> respectively, then the sum of the squares of distances, is as</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mfenced><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi></mrow><msqrt><mn>3</mn></msqrt></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mfrac><mrow><mi>x</mi><mo>-</mo><mi>z</mi></mrow><msqrt><mn>2</mn></msqrt></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mfrac><mrow><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi></mrow><msqrt><mn>6</mn></msqrt></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>p</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>z</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>6</mn><msup><mi>p</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>  </mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>4</mn><mi>y</mi><mi>z</mi><mo>+</mo><mn>4</mn><mi>z</mi><mi>x</mi><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mi>z</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>x</mi><mi>y</mi><mo>-</mo><mn>4</mn><mi>y</mi><mi>z</mi><mo>+</mo><mn>2</mn><mi>x</mi><mi>z</mi><mo>=</mo><mn>6</mn><msup><mi>p</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>y</mi><mo>+</mo><mn>6</mn><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><mn>6</mn><msup><mi>p</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><msup><mi>p</mi><mn>2</mn></msup></math></p><p>Hence, option (d) is correct.</p></div>


tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
if-the-sum-of-the-coefficients-in-the-expansion-of-x-y-n-is-1024-then-the-value-of-the-greatest-coefficient-in-the-expansion-is

slug:
if-the-sum-of-the-coefficients-in-the-expansion-of-x-y-n-is-1024-then-the-value-of-the-greatest-coefficient-in-the-expansion-is-42142

question:
<div class="question">If the sum of the coefficients in the expansion of $(x+y)^n$ is 1024 , then the value of the greatest coefficient in the expansion is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$356$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$252$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$210$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$120$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$252$
<br/></span> </div>

answer:
<div class="solution">Given, sum of the coefficient $=1024$
<br/>$\begin{array}{ll}\text { i.e. } &amp; 2^n=1024=2^{10} \\ \Rightarrow &amp; n=10\end{array}$
<br/>Since, $n$ is even, so greatest coefficient
<br/>$={ }^n C_{n / 2}={ }^{10} C_5=252$</div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">11</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-sum-of-the-first-2-n-terms-of-258-ldots-ldots-is-equal-to-the-sum-of-the-first-n-terms-of-5759-61-ldots-ldots-then-mathrmn-is-equal-to

slug:
if-the-sum-of-the-first-2-n-terms-of-258-ldots-ldots-is-equal-to-the-sum-of-the-first-n-terms-of-5759-61-ldots-ldots-then-mathrmn-is-equal-to-82269

question:
<div class="question">If the sum of the first \(2 n\) terms of \(2,5,8, \ldots \ldots\) is equal to the sum of the first \(n\) terms of 57,59 , \(61 \ldots \ldots\). , then \(\mathrm{n}\) is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">11</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">13</span> </li> </ul>

answer:
<div class="solution">Given,<br/>\(\frac{2 n}{2}\{2.2+(2 n-1) 3\}=\frac{n}{2}\{2.57+(n-1) 2\}\)<br/>or \(2(6 n+1)=112+2 n\) or \(10 n=110, ~\therefore n=11\)</div>


title: if-the-tangent-at-p-1-1-on-y-2-x-2-x-2-meets-the-curve-again-at-q-then-q-is
slug: if-the-tangent-at-p-1-1-on-y-2-x-2-x-2-meets-the-curve-again-at-q-then-q-is-33803
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the tangent at $P(1,1)$ on $y^{2}=x(2-x)^{2}$ meets the curve again at $Q$, then $Q$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(2,2)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(-1,-2)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(\frac{9}{4}, \frac{3}{8}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{9}{4}, \frac{3}{8}\right)$</span> </div>

answer:
<div class="solution">$\begin{array}{l}
<br/> <br/>y^{2}=x(2-x)^{2} \Rightarrow y^{2}=x^{3}-4 x^{2}+4 x \ldots \text { (i) } \\
<br/> <br/>\Rightarrow 2 y \frac{\mathrm{dy}}{\mathrm{dx}}=3 x^{2}-8 x+4 \\
<br/> <br/>\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{3 x^{2}-8 x+4}{2 y} \\
<br/> <br/>\Rightarrow\left[\frac{\mathrm{dy}}{\mathrm{dx}}\right]_{\mathrm{P}}=\frac{3-8+4}{2}=-\frac{1}{2}
<br/> <br/>\end{array}$
<br/> <br/>$y^{2}=x(2-x)^{2} \Rightarrow y^{2}=x^{3}-4 x^{2}+4 x \ldots$ (i) $\Rightarrow 2 y \frac{\mathrm{dy}}{\mathrm{dx}}=3 x^{2}-8 x+4$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{3 x^{2}-8 x+4}{2 y}$ $\Rightarrow\left[\frac{\mathrm{dy}}{\mathrm{dx}}\right]_{\mathrm{P}}=\frac{3-8+4}{2}=-\frac{1}{2}$ $\therefore$ Equation of tangent at $\mathrm{P}$ is: $y-1=-\frac{1}{2}(x-1)$ $\Rightarrow x+2 y-3=0$ $\Rightarrow x+2 y-3=0$
<br/> <br/>Using $y=\frac{3-x}{2}$ in (i), we get: $\left(\frac{3-x}{2}\right)^{2}$
<br/> <br/>$\begin{array}{l}
<br/> <br/>=x^{3}-4 x^{2}+4 x \\
<br/> <br/>\Rightarrow 4 x^{3}-17 x^{2}+22 x-9=0 \text {...(ii) }
<br/> <br/>\end{array}$
<br/> <br/>which has two roots 1,1 (Because of (ii) being tangent at $(1,1)$ ).
<br/> <br/>Sum of 3 roots $=\frac{17}{4}$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\therefore 3 \mathrm{rd} \text { root }=\frac{17}{4}-2=\frac{9}{4} \\
<br/> <br/>\text { Then, } y=\frac{3-\frac{9}{4}}{2}=\frac{3}{8} \\
<br/> <br/>\therefore \mathrm{Q} \text { is }\left(\frac{9}{4}, \frac{3}{8}\right)
<br/> <br/>\end{array}$</div>


title: if-the-the-earth-is-at-onefourth-of-its-present-distance-from-the-sun-the-duration-of-year-will-be
tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2020']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P1.db
slug:
if-the-the-earth-is-at-onefourth-of-its-present-distance-from-the-sun-the-duration-of-year-will-be-82327

question:
<div class="question">If the the earth is at one-fourth of its present distance from the sun, the duration of year will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">half the present year</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">one-eight the present year</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">one-fourth the present year</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">one-sixth the present year</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">one-eight the present year</span> </div>


title: if-the-third-term-in-the-expansion-of-x-x-l-o-g-10-x-5-is-1-0-6-then-x-may-be
slug: if-the-third-term-in-the-expansion-of-x-x-l-o-g-10-x-5-is-1-0-6-then-x-may-be-66334
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the third term in the expansion of $\left[\mathrm{x}+\mathrm{x}^{\log _{10} \mathrm{x}}\right]^{5}$
<br/> <br/>is $10^{6},$ then $\mathrm{x}$ may be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{10}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">10</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$10^{-2 / 5}$</span> </li> </ul>

answer:
<div class="solution">Put $\log _{10} x=y,$ the given expression becomes $\left(x+x^{y}\right)^{5}$
<br/> <br/>$\mathrm{T}_{3}={ }^{5} \mathrm{C}_{2} \cdot \mathrm{x}^{3}\left(\mathrm{x}^{\mathrm{y}}\right)^{2}=10 \mathrm{x}^{3+2 \mathrm{y}}=10^{6}$ (given)
<br/> <br/>$\Rightarrow(3+2 \mathrm{y}) \log _{10} \mathrm{x}=5 \log _{10} 10=5$
<br/> <br/>$\Rightarrow(3+2 \mathrm{y}) \mathrm{y}=5$
<br/> <br/>$\Rightarrow \mathrm{y}=1,-\frac{5}{2}$
<br/> <br/>$\Rightarrow \log _{10} \mathrm{x}=1$ or $\log _{10} \mathrm{x}=-\frac{5}{2}$
<br/> <br/>$\therefore \mathrm{x}=10$ or $\mathrm{x}=(10)^{-5 / 2}$</div>


tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
if-the-time-period-is-doubled-then-the-angular-momentum-of-the-body-will-provided-the-moment-of-inertia-of-the-body-is-constant

slug:
if-the-time-period-is-doubled-then-the-angular-momentum-of-the-body-will-provided-the-moment-of-inertia-of-the-body-is-constant-69789

question:
<div class="question">If the time period is doubled, then the angular momentum of the body will (provided the moment of inertia of the body is constant)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">remain constant</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">quadruple</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">become half</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">double</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">become half</span> </div>

answer:
<div class="solution"><p>As we know.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mi>I</mi><mi>ω</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>L</mi><mo>=</mo><mi>I</mi><mo>×</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>T</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>∝</mo><mfrac><mn>1</mn><mi>T</mi></mfrac></math></p><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mi>T</mi></math></p><p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>L</mi><mn>1</mn></msub><msub><mi>L</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>T</mi></mrow><mi>T</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msub><mi>L</mi><mn>2</mn></msub><mo>=</mo><mfrac><msub><mi>L</mi><mn>1</mn></msub><mn>2</mn></mfrac></math></p></div>


tags: ['Physics', 'Nuclear Physics', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
title:
if-the-total-binding-energies-of-1-2-h-2-4-he-26-56-fe-92-235-u-nuclei-are-222283492-and-1786-mev-respectively-identify-the-most-stable-nucleus-of-the

slug:
if-the-total-binding-energies-of-1-2-h-2-4-he-26-56-fe-92-235-u-nuclei-are-222283492-and-1786-mev-respectively-identify-the-most-stable-nucleus-of-the-61499

question:
<div class="question">If the total binding energies of ${ }_{1}^{2} \mathrm{H},{ }_{2}^{4} \mathrm{He},$ ${ }_{26}^{56} \mathrm{Fe} \&amp;{ }_{92}^{235} \mathrm{U}$ nuclei are 2.22,28.3,492 and 1786 MeV respectively, identify the most stable nucleus of the following.</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">${ }_{26}^{56} \mathrm{Fe}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">${ }_{1}^{2} \mathrm{H}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">${ }_{92}^{235} \mathrm{U}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">${ }_{2}^{4} \mathrm{He}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">${ }_{26}^{56} \mathrm{Fe}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \mathrm{B.E}_{\mathrm{H}} &amp;=\frac{2.22}{2}=1.11 \\ \mathrm{B.E}_{\mathrm{He}} &amp;=\frac{28.3}{4}=7.08 \\ \mathrm{B.E}_{\mathrm{Fe}} &amp;=\frac{492}{56}=8.78=\text { maximum } \\ \mathrm{B.E}_{\mathrm{U}} &amp;=\frac{1786}{235}=7.6 \end{aligned}$
<br/>$\frac{56}{26} \mathrm{Fe}$ is most stable as it has maximum binding energy per nucleon.</div>


tags: ['Physics', 'Work Power Energy', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
title:
if-the-unit-of-force-and-length-be-each-increased-by-four-times-then-the-unit-of-energy-is-increased-by

slug:
if-the-unit-of-force-and-length-be-each-increased-by-four-times-then-the-unit-of-energy-is-increased-by-80361

question:
<div class="question">If the unit of force and length be each increased by four times, then the unit of energy is increased by</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">16 times</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">8 times</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2 times</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4 times</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">16 times</span> </div>

answer:
<div class="solution">Since unit of energy $=$ (unit of force). (unit of length) so if we increase unit of length and force, each by four times, then unit of energy will increase by sixteen times.</div>


title: if-the-value-of-mode-and-mean-is-60-and-66-respectively-then-the-value-of-median-is
slug: if-the-value-of-mode-and-mean-is-60-and-66-respectively-then-the-value-of-median-is-65445
tags: ['Mathematics', 'Statistics', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">64</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the value of mode and mean is 60 and 66 respectively, then the value of median is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">70</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">64</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">90</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">50</span> </li> </ul>

answer:
<div class="solution">$\quad$ Mode $=3$ Median -2 Mean
<br/> <br/>$\therefore \quad$ Median $=\frac{1}{3}($ mode +2 mean $)=\frac{1}{3}(60+2 \times 66)=64$</div>


title: if-the-variance-of-the-observations-x-1-x-2-x-n-is-2-then-the-variance-of-x-1-x-2-x-n-0-is
slug: if-the-variance-of-the-observations-x-1-x-2-x-n-is-2-then-the-variance-of-x-1-x-2-x-n-0-is-46618
tags: ['Mathematics', 'Statistics', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the variance of the observations <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mn>2</mn></msup></math>, then the variance of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mi>α</mi><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>…</mo><mo>,</mo><mi>α</mi><msub><mi>x</mi><mi>n</mi></msub><mo>,</mo><mi>α</mi><mo>≠</mo><mn>0</mn></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><msup><mi>σ</mi><mn>2</mn></msup></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>α</mi><mn>2</mn></msup><msup><mi>σ</mi><mn>2</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>σ</mi><mn>2</mn></msup><msup><mi>α</mi><mn>2</mn></msup></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>α</mi><mn>2</mn></msup><msup><mi>σ</mi><mn>2</mn></msup></math></span> </div>

answer:
<div class="solution"><p>Variance <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mi>Σ</mi><mo>(</mo><mi>x</mi><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover><mrow><msup><mo>)</mo><mn>2</mn></msup><mo>=</mo></mrow><msup><mi>σ</mi><mn>2</mn></msup></math></p><p>New variance <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mi>Σ</mi><mo>(</mo><mi>α</mi><mi>x</mi><mo>-</mo><mi>α</mi><mover><mi>x</mi><mo>¯</mo></mover><msup><mo>)</mo><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mfrac><mn>1</mn><mi>n</mi></mfrac><mi>Σ</mi><mo>(</mo><mi>x</mi><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover><msup><mo>)</mo><mn>2</mn></msup><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><msup><mi>σ</mi><mn>2</mn></msup></math></p></div>


tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
title:
if-three-vertices-of-a-regular-hexagon-are-chosen-at-random-then-the-chance-that-they-form-an-equilateral-triangle-is

slug:
if-three-vertices-of-a-regular-hexagon-are-chosen-at-random-then-the-chance-that-they-form-an-equilateral-triangle-is-58951

question:
<div class="question">If three vertices of a regular hexagon are chosen at random, then the chance that they form an equilateral triangle is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{5}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{10}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{10}$</span> </div>

answer:
<div class="solution">Three vertices can be selected in ${ }^{6} \mathrm{C}_{3}$ ways.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/YwWmXDaEV9LZX_I_avY1qkPu_rA3XsvxmHeX49ysZMc.original.fullsize.png"/><br/> <br/> <br/>The only equilateral triangles possible are $\mathrm{A}_{1} \mathrm{~A}_{3} \mathrm{~A}_{5}$ and $\mathrm{A}_{2} \mathrm{~A}_{4} \mathrm{~A}_{6}$
<br/> <br/>$\mathrm{p}=\frac{2}{{ }^{6} \mathrm{C}_{3}}=\frac{2}{20}=\frac{1}{10}$</div>


tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
title:
if-two-distinct-chords-drawn-from-the-point-p-q-on-the-circle-x-2-y-2-p-x-q-y-where-p-q-0-are-bisected-by-the-x-axis-then

slug:
if-two-distinct-chords-drawn-from-the-point-p-q-on-the-circle-x-2-y-2-p-x-q-y-where-p-q-0-are-bisected-by-the-x-axis-then-45560

question:
<div class="question">If two distinct chords drawn from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math> on the circle <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mi>p</mi><mi>x</mi><mo>+</mo><mi>q</mi><mi>y</mi></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>q</mi><mo>≠</mo><mn>0</mn></math>) are bisected by the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>-axis, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><msup><mi>q</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><mn>8</mn><msup><mi>q</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>&lt;</mo><mn>8</mn><msup><mi>q</mi><mn>2</mn></msup></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>&gt;</mo><mn>8</mn><msup><mi>q</mi><mn>2</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>&gt;</mo><mn>8</mn><msup><mi>q</mi><mn>2</mn></msup></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>t</mi><mo>,</mo><mo> </mo><mi>m</mi><mo>)</mo></math> be the other end of the chord drawn from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>)</mo></math> on the circle</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mi>p</mi><mi>x</mi><mo>+</mo><mi>q</mi><mi>y</mi></math></p><p>Their mid-point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mrow><mi>t</mi><mo>+</mo><mi>p</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mfrac><mrow><mi>m</mi><mo>+</mo><mi>q</mi></mrow><mn>2</mn></mfrac></mrow></mfenced></math></p><p>since, mid-point lies on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>-axis i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>+</mo><mi>q</mi><mo>=</mo><mn>0</mn></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math></p><p>Also, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>t</mi><mo>,</mo><mo> </mo><mi>m</mi><mo>)</mo></math> lies on the circle.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mi>p</mi><mi>t</mi><mo>-</mo><mi>q</mi><mi>m</mi><mo>=</mo><mn>0</mn></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>ii</mi><mo>)</mo></math></p><p>From Eqs. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>ii</mi><mo>)</mo></math>, we get <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mi>p</mi><mi>t</mi><mo>+</mo><mn>2</mn><msup><mi>q</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math></p><p>Which is quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> such that, Discriminant <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&gt;</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><msup><mi>q</mi><mn>2</mn></msup><mo>&gt;</mo><mn>0</mn><mo>⇒</mo><msup><mi>p</mi><mn>2</mn></msup><mo>&gt;</mo><mn>8</mn><msup><mi>q</mi><mn>2</mn></msup></math></p></div>


